What Adjusted Plus-Minus (APM) in brief is:
Over a given time period, the basic plus-minus results are getting adjusted to account for both the teammates and the opponents on the court.

What Adjusted Plus-Minus does:
It reflects the impact of each player on his team's scoring margin after controlling for the strength of every teammate and every opponent during each minute he's on the court.

What Adjusted Plus-Minus incorporates:
Every time segment a player is in a game, adjusted plus-minus tracks:
(1) The other nine players on the floor,
(2) The length of the segment,
(3) The score at the start and at the end of the segment.

How to interpret Adjusted Plus-Minus numbers:
Adjusted +/- ratings indicate how many additional points are contributed to a team’s scoring margin by a given player in comparison to the league-average player whose adjusted +/- value is zero over the span of a typical game. It is assumpted that in a typical game a team has 100 offensive and 100 defensive possessions. For example, if a +6.5 "adjusted +/-" player is on the floor with 4 average teammates, his team will average about 6.5 points better per 100 possessions than 5 average players would.

How are the estimates for Adjusted Plus-Minus calculated?
It's a matter of finding out the estimates of player variables which produces the smallest difference between expected margin and the actual margin in the matchups. This is how a regression model works, basically.
Get more information about adjusted plus-minus calculation

What are the PROs and CONs for Adjusted Plus-Minus?
PROS:
The biggest advantage of adjusted plus/minus ratings are that they are one of the the closest we can come to an unbiased measure of a player's effectiveness.
CONS:
(1) Adjusted plus/minus ratings are have a high variance. The regression tries to find a constant value for a player, but this value can change pretty dramatically with a different role, a different coaching scheme, different teammates, or different match-ups.
(2) There is noise in the data. For some players, especially when only looking at data over just 1 year, there are some strange results, but that is to be expected. An examination of 239 players revealed that only 7% of the variation in a player’s adjusted plus–minus value in 2008-09 was explained by what he did in 2007-08. Although more data does increase the level of statistical significance, it’s still the case that most players even when five years of data is employed are not found by this method to have a statistically significant impact.
(3) Another issue which adjusted plus-minus technique struggles to address is the multicollinearity issue. Coaches prefer to use some player duos/trios frequenly or rarely since all players could not be on the court with every other teammate at the same time.

History of Adjusted Plus-Minus Development:
Adjusted plus/minus technique is first developed by Wayne Winston and Jeff Sagarin in the form of their WINVAL system in 2002. For each player, it starts with the team’s average point differential for each possession when they are on the court. This gives a number showing how effective the player’s team was when they were in the game. The problem with using this to evaluate individual players is that it is biased in favor of players who play alongside great teammates and players who play against weak opponents.
Dan Rosenbaum initially outlined his approach in an influential article that analyzed the 2002-2003 and 2003-2004 seasons as a combined dataset.
David Lewin then published both 2004-2005 and 2005-2006 ratings.
Likewise, Steve Ilardi determined adjusted +/- ratings for the 2006-2007 season using a similar approach. While the previous analyses were conducted at the conclusion of a season, he presented the first in-season computation of adjusted +/-.
Since 2007-08 season, daily updates of the APM calculations are available on Aaron Barzilai's BasketballValue website.

The metric which is an estimate of a player's value, making no fine distinctions, but, rather, distinguishing easily between very good seasons, average seasons, and poor seasons. There are two ways to calculate approximate value:

Credits=(Points)+(Rebounds)+(Assists)+(Steal)+(Blocks)-(Field Goals Missed)-(Free Throws Missed)-(Turnovers)
AV= (Credits^(3/4) )/21

Alternative method shall be assigned according to the following rules shown here

Assist percentage is an estimated percentage of teammate field goals a player assisted while the player is on the court.

Assist Percentage=100*Assists/(((Minutes Played /(Team Minutes/5)) * Team Field Goals Made) - Field Goals Made)

A metric that measures the difference per 100 possessions in points allowed with a player on the court versus off the court. The accuracy of this measure varies dependent upon how often the player is on the floor and whom the player share it with. In most cases it provides a good indication of a player’s overall defensive value to his team.

A metric which is invented by Dean Oliver. The premise of individual defense is that players force "defensive stops", preventing the other team from scoring.

An individual can do that by forcing a missed shot that then gets rebounded by his team, by getting a defensive rebound, by forcing a turnover, or by fouling a player who misses both foul shots, the second of which is then rebounded by the defense.

A defensive stop occurs when a player or team defense regains the ball without allowing the opponent a scoring possession.

Ways to get the "defensive stops" done:
1. Forcing a missed shot that then gets rebounded by his team,
2. Getting a defensive rebound,
3. Forcing a turnover,
4. Fouling a player who misses both foul shots, the second of which is then rebounded by the defense.

A metric which is invented by Kevin Broom who devised a simple general formula that works with any per-minute statistics. Fomula subtracts the player's rating per game from his rating per 40 minutes to figure out how much his per game stats undervalue his potential contributions. He then subtracts league average from the player's per-40 minute rating and adds this amount to ensure the player is actually playing well in the minutes he does get.

Diamond Rating Formula=(Per Minute Rating)*40-(Per Minute Rating)*(Minutes Per Game)+[(Per Minute Rating)-(League's Per Minute Rating)]*40

A metric which is John Hollinger’s simple and linear version of the Player Efficiency Rating.

Game Score Formula =(Points)+0.4*(Field Goals Made)+0.7*(Offensive Rebounds)+0.3*(Defensive rebounds)+(Steals)+0.7*(Assists)+0.7*(Blocked Shots)- 0.7*(Field Goal Attempts)-0.4*(Free Throws Missed) - 0.4*(Personal Fouls)-(Turnovers)

The metric that indicates the ratio of a player's scoring possessions by his total possessions. 

When a player ends his team's possession, it would be a possession charged to him. This is player's total possessions. When a player scored or assisted on a score, a scoring possession would be charged to him.

Individual Floor Percentage Formula=100*(Player's Scoring Possessions)/(Player's Total Possessions)

A linear and basic metric which indicates players' efficiency.

NBA Efficiency Formula = (Points)+(Rebounds)+(Steals)+(Assists)+(Blocked Shots)-(Turnovers)-(Missed Shots)

NBA.com Plus-Minus is a plus/minus tool to find out best player combinations. It looks at the point differential when players are both in and out of the game, to see how the team performs with various combinations.

A variety of combinations including the best two player, three player and even five player combinations for each game can be provided with this tool.

Net plus-minus, a.k.a. unadjusted plus-minus, is invented by Roland Beech and measure the plus-minus statistics for a given player when the player is in the game relative to the plus-minus statistics when the player is NOT in the game. Net plus-minus combines offensive and defensive plus-minus.

For example, if a team scores 115 points (per 100 possessions) while a given player is on the court and 98 points (per 100 possessions) while he is off the court, his net offensive plus-minus is +17 (115-98=17).
If the team allows 110 points (per 100 possessions) while given player is on the court and 105 points (per 100 possessions) while he is off the court, his net defensive plus-minus is +5 (110-105=+5).
So, net plus-minus for the given player is +17-(+5)=+12

Net Points is one way to burn indivudual Offensive and Defensive ratings into one metric.

Net Points=(Points Produced)-(Points Allowed)
Where "Points Allowed" can be calculated by using the following formula:
Points Allowed=[(DefensiveRating/100)*(.2*{(MinutesPlayed/(TeamMinutes/5)}*TeamPossession))]

Offensive Conversion Rate measures the percentage of a player's steals that result in a made basket or free throw attempts within 5 seconds.

The reason for taking 5 seconds is to measure a player's transition defense in terms of how frequently his steals lead to offensive production.

A metric which is invented by Dean Oliver. Individual offensive ratings are constructed from the following two statistics:

• Individual possessions, which represents the credit an individual gets for the times his/her team ends a possession and gives it back to the opponents.

• Individual points produced, which represents the credit an individual gets for the points that his/her team generates on the offensive end.

The metrics that evaluates a player's performance more accurately than his averages.

The common notation for per-minute ratings is using per 40 minute stats.

Per minutes ratings are calculated by taking the player's total in any category (points, rebounds, assists, etc.) divided by total minutes played.

Personal Foul Efficiency measures ratio of steals plus blocks to personal fouls. Indicates how efficiently a player forces turnovers, removing the bias of jump-happy or slaptastic players who accumulate deceptive block or steal totals.

The metric that which boils down all of a player's contributions into one number. John Hollinger's formula adds positive stats and subtracts negative ones through a statistical point value system.

For more information about PER Calculation visit basketball-reference

A player's non-scoring possessions would be his missed field goals, plus free throws that weren't rebounded by his team, plus his turnovers.

Non-Scoring Possessions Formula=(Player's Field Goal Attempts)-Player's Field Goal Made)+0.4*(Free Throw Attempts)+(Player's Turnovers)

A player's scoring possessions would be his field goals that weren't assisted on, plus a certain percentage of his field goals that were assisted on, plus a certain percentage of his assists, plus his free throws made.

Scoring Possessions Formula=(Field Goals Made) - 0.37*(Field Goals Made)*Q/R + 0.37*(Player Assists) + 0.5*(Free Throws Made)

where;

Q=5*(Player Minutes)*(Team Assist Total)/(Team Total Minutes)-(Player Assists) 
R=5*(Player Minutes)*(Team Field Goals Made)/(Team Minutes)-(Player Assists)

The metric that represents a player's total possessions in the game. In other words,a player's total possessions is the total number of how many times he ends his team's possession.

Player's Total Possessions Formula=(Player's Scoring Possessions)+(Player's Non-Scoring Possessions)=(Player Field Goal Attempts)-(Player Missed Shots)*(Team Offensive Rebounding Percentage)+0.37*(Player Assists)-0.37*(Field Goals Made)*Q/R+(Player Turnovers)+0.4*(Player Free Throw Attempts)

Where;

Q=5*(Player Minutes)*(Team Assist Total)/(Team Total Minutes)-(Player Assists)
R=5*(Player Minutes)*(Team Field Goals Made)/(Team Minutes)-(Player Assists)
 

The plus-minus, a.k.a. +/-, simply keeps track of the net changes in score when a given player is either on or off the court.

NBA.com has been publishing plus-minus values in official box scores since 2007-2008 season.

Since the invention of basic plus minus, the plus-minus technique has been developed in order to get more accurate results. At the moment, there are 4 advanced plus minus metrics that derived from basic one.

• Net Plus-Minus (NPM)
• Adjusted Plus-Minus (APM)
• Statistical Plus-Minus (SPM)
• Regularized Adjusted Plus-Minus (RAPM)

Points created is a metric that evaluates overall player performance.

It comprises all current primary and secondary statistical categories: points, rebounds, assists, steals, blocked shots, missed shots (both field goals and free throws), turnovers, and personal fouls.

Points Created= PTS + AST * (2-VBP) + (REB + STL + BLK) * VBP -(FGMiss + FTMiss + TOV) * VBP – 0.5 * VBP * PF
VBP = the value of ball possession is league’s average points per 100 possessions.

It measures overall performance rather than a single skill (such as shooting) or set of skills (such as shooting, rebounding, and shot-blocking). Therefore, the result is a clear composite of a team or player's total performance.

Points Produced is a measure of player's offensive contribution. "Points produced" metric has been considered more important than the "points scored".

A player produces points through made shots, assists, and offensive rebounds. The number of points produced by the various methods depends on the actual number of points scored and the importance of the player's contribution.

The "Win Score" metric that takes player's "primary position played" into account.

PAWS=Position Adjusted Win Score=[(PAWS48/48)*Minutes]

PAWS48 =PAWS per 48 minutes=[WS48 – Average WS48 at primary position played]

WS48 =Win Score per 48 minutes

Total rebound percentage is estimated percentage of available rebounds grabbed by the player while the player is on the court.

Total Rebound Percentage=100*(Total Rebounds*(Team Minutes Played/5))/(Minutes Played*(Team Total Rebounds + Opponent Team's Total Rebounds))

In "Regularized Adjusted Plus-Minus" (RAPM), the goal is to provide more accurate results by employing a special technique called "ridge regression" (a.k.a. regularization). It significantly reduces standard errors in Adjusted Plus-Minus (APM).

Conventional adjusted plus-minus is shown to do a poor job of predicting the outcome of future games, particularly when fit on less than one season of data. Adding regularization greatly improves accuracy, and some player ratings change dramatically. The enhancement with the RAPM is a Bayesian technique in which the data is combined with a priori beliefs regarding reasonable ranges for the parameters in order to produce more accurate models. That is what ridge regression (a.k.a. regularization) does.

The Roland Rating is a basic, quick look at the player that represent a player's value to a particular team and are not intended to be an accurate gauge of the ability and talent of the player away from the specific team. It represents how successful a player is with a given team.

The on court +/- number represents the team's net points with the player on the floor per 48 minutes, while the off court number is the team's net with the player off the floor per 48 minutes. The Roland Rating is the difference between the two, with a positive number indicating the team has played better with the player than without.

The estimate of how many seasons the player has left to play.

Seasons Left Formula=27-0.75*(Age of The Player) 

It's been developed to account for aging when calculating player efficiency. SPS uses 3 years of data, with the most recent data weighted heavier. It regresses towards the mean. So the model has an age factor and the numbers are adjusted to account for age.

An example of how the SPS is calculated for Dwight Howard's rebounding in 2008-09 season by using data of previous three seasons.

Statistical Plus/Minus (SPM) is an estimate of the player's contribution to the team's point differential per 100 possessions, using his box score stats as inputs. In another words, SPM can be considered as a way of estimating adjusted plus-minus from box score stats. Statistical plus minus puts a weight to basic box score stats like, points, steals, blocks etc., in a effort to solve the lowest mean residuals (average errors).

The goal with the SPM technique is adding a more stable component to player's basic stats as a counterbalance to the basic plus-minus. As a well known fact, basic plus-minus can be wildly inconsistent for individual players from season to season or under different coaches in a season.

Dan Rosenbaum has outlined different ways of addressing adjusted plus-minus' accuracy issues, by employing additional analyses based on box score stata.

In a lot of cases the statistical plus/minus ratings might be a more accurate predictor of future defensive performance. To characterize past defensive performance, adjusted plus-minus does a good job. A combination of the adjusted and statistical plus-minus will do a better job when we want to find out which players are going to be good or bad defenders in the future.

Steal Percentage is the percentage of estimated opponent possessions that end with a steal by the player while the player is on the court.

Steal Percentage=100*(Steals*(Team Minuted Played/5))/(Minuted Played*Opponent's Possessions)

Dave Heeren's player evaluation metric which is generally considered the first player rating system with linear weights.

The Tendex Formula=[(Points)+(Rebounds)+(Assists)+(Steals)+(Blocks)- (Missed Field Goal Attempts)-0.5*(Missed Free Throws)-(Turnovers)-(Fouls)]/(Player's Minutes)

Touches estimate the number of times a player touched the ball in an attacking position on the floor. The theory behind the formula is that once a player gets the ball, he can only do one of four things (aside from dribbling, of course): pass, shoot, draw a foul, or commit a turnover.

Touches=Field Goal Attempts + Turnovers + (Free Throw Attempts / (Team's Free Throw Attempts/Opponents Personal Fouls)) + (Assists/0.17)

• %Pass = 100 * (AST / 0.17) / Touches. The percentage of a player's touches that ended with him passing to a teammate in an attacking position on the floor.
• %Shoot = 100 * FGA / Touches. The percentage of a player's touches that ended with him taking a shot and not being fouled in the act.
• %Fouled= 100 * (FTA / (Tm FTA / Opp PF)) / Touches. The percentage of a player's touches that ended with him drawing a foul.
• %TO = 100 * Turnovers / Touches. The percentage of a player's touches that ended with him turning the ball over.

The estimate using a player's age and his approximate value to determine how much value a player has left in his career.

Trade Value Formula=[(Approximate Value- 27-0.75*Age )^2( 27-0.75*Age +1)*Approximate Value]/190+(Approximate Value)*2/13

The mathematical relationship between usage rate and efficiency.

Trade off between usage and efficiency explains the player's gain of offensive rating (points per 100 possessions) for drop of each percent of usage.

The metric that factors a player's performance at the free-throw line and considers their efficiency on all types of shots. This data will fill any empty slots on your stat sheets.

True Shooting Percentage Formula=(Player's Total Points)/[(2*(Player's Field Goal Attempts+ 0.44*Player's Free Throw Attempts)]

Usage percentage is an estimate of the percentage of team plays used by a player while he was on the floor.

Usage Percentage Formula=100*[(Team Minutes)/(5*(Player Minutes))]*[(Field Goal Attempts)+0.44*(Free Throw Attempts)+(Turnovers)/[(Team Field Goal Attempts)+0.44*(Team Free Throw Attempts)+Team Turnovers)]

By balancing usage rates and the varying offensive ratings of the five players on the court, a team can achieve optimal offensive output for the personnel.

The Versatility Index, which was designed by John Hollinger, is a metric that measures a player's ability to produce in more than one statistic. The metric uses points, assists, and rebounds. The average player will score around a five on the index, while top players score above 10. and more information can be found in this article.

Versatility Index Formula=[(PPG)*(RPG)*APG)]^(0.333)

A program which measures team chemisty and analyzes a player's impact on his team's ability to produce points. WINVAL is the labor of Jeff Sagarin and Wayne Winston, who are both graduates of MIT.

WINVAL's most unique characteristic is that it measures individual and team performance after every lineup change in every game. WINVAL's ratings also factor in who each individual is playing with and against at all times. The WINVAL scouting reports also include data on referees.

The metric that indicates the relative value of a player's point, rebound, steal, turnover, and field goal attempt.

Win Score Formula=(Points)+(Rebounds)+(Steals)+(½Assists)+(½Blocked Shots)-(Field Goal Attempts)-(Turnovers)-½(Free Throw Attempts)-½Personal Fouls

A Win Share is worth one-third of a team win. Win Shares are assigned to players based on their offense, defense, and playing time. If a team wins 60 games, there are 180 Win Shares to distribute among the players. This is always true; if a team wins n games, then there are 3n Win Shares to allocate to the players.

More information about Win Share calculation

WARP, a metric which is invented by Kevin Pelton, stands for Wins Above Replacement Player. The term and concept are borrowed from sabermetrics and, specifically, Baseball Prospectus.

Conceptually, the WARP system seeks to evaluate players in the context of a team made up of them and four completely average players. The performance of this team is then compared to that of a team made up of four average players and one replacement level player. The method also draws heavily on the work of Dean Oliver.

Using replacement level shows the value of players who can play heavy minutes and avoid injury while continuing to perform above replacement level. Using wins gives a measure of value that is easy to understand and constant over time. Lastly, by eschewing the traditional linear weights method so common in basketball analysis, WARP does a better job of incorporating defensive value. Learn more about WARP calculation.

A metric which is invented by David J. Berri based on box score stats. Wins Produced is derived from the relationship between wins and offensive and defensive efficiency. It takes five steps to calculate Wins Produces for each player.

Get more information about Wins Produced calculation.