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Each of the pieces in chess has a different value indicating its usefulness relative to other pieces. Under most systems, including that used in the "Details" section of Chess.com games, a pawn is worth one point, a knight is worth three points, a bishop three points, a rook five points, and the queen nine points. One can use this system to calculate the merit of an exchange--for instance, two knights (six points) for a rook and two pawns (seven points) is generally a good exchange for the player losing the knights.
Assigning value to the king is more difficult, since its "loss" (i.e. checkmate) is the end of the game. Thus it is often said that the king is worth an infinte number of points. However, a piece's value is normally intended to reflect its usefulness in attack and defense. In this scheme, the king is usually said to be worth about three points.
Of course, these values are not absolute. The pieces are not everything in chess--their position matters too. Thus, a player may sacrifice a piece for a positional advantage (as is the case in the King's Gambit Accepted, 1. e4 e5 2. f4 exf4, in which White is down a pawn, but has diverted Black's e-pawn to the outside, thus improving White's chances of center control) or to open up a mating attack (see, for instance, the end of the Immortal Game, in which the Queen and two rooks are sacrificed).
The values of pieces may also change based on their position. Passed pawns (those with no pawns able to block their advance to promotion) are worth more than unpassed pawns, because they constitute a greater threat to the opponent; as they advance towards the promotion square, they become worth more and more.
Pieces also change value somewhat in the endgame. Two bishops (six points) can deliver checkmate against a lone king, while two knights cannot unless the opponent blunders. (This is one of the reasons some say the bishop is worth a bit more than the knight).
Many chess engines use an adjusted piece value system (which incorporates positional advantages and such) to evaluate positions. Little ChessPartner, Chess.com's computer opponent, analyzes advantage in centipawns (hundredths of a pawn), giving an estimate of advantage after each turn.