A chess rating is a number that, in essence, provides a numerical comparison between a particular player and other chess players. The number is based upon the player's performance in tournaments based upon the performance against a sum of each specific opponent and that opponent's rating. This system, in the end, allows players to judge their chess strength relative to other rated chess players.
The method of chess ratings is very general and can also be applied to other games or sports in which there are a large number of individuals competing. Besides chess, for instance, Magic: the Gathering uses a nearly identical system. Having each player play each other player under a variety of conditions is logistically impossible, so in order to have a fair way to grade players, a numerical scheme that constantly compares player performances is necessary, and thus the chess rating system was developed by FIDE and USCF.
The chess rating system is calculated as follows. When a player first becomes registered with a governing body of the game, he or she is assigned a base rating of 1600. This is considered to be the average score of all players, so you start off at the fiftieth percentile among all rated players (in theory; in practice, many players with lower ratings quit the game, so among active players, your percentile is likely much lower).
Once you have your rating and the rating of your opponent, you then calculate your win expectancy (We) against that opponent using the following formula:
1 We = win expectancy
We = -------------------- R1 = rating of the player
10^((R2-R1)/400) + 1 R2 = rating of the opponent
Given the win expectancy, then the result of the match is taken into consideration, and each player is awarded a score. A win gets a score of 1, a draw gets a score of 0.5, and a loss gets a score of 0. Also, each match is given a particular K value. K is a number that tells how many points are at stake in a given match; more important matches have a higher K value. Common K values are 24 and 32 for most low-level tournaments.
We then use the score in the match (W), the old rating (Ro), the K value, and the win expectancy (We) and combine them all together to calculate the new rating (Rn), as shown below:
Rn = Ro + (K * (W - We))
The rating is recalculated at the end of each match.
Let's take an example. Let's say we have a new player, Bobby. He's given a rating of 1600, and his first match is against Bill, who has a rating of 1700, in a tournament with a K value of 24. Let's calculate the We:
1
We = ------------------------ = 0.359935
10^((1700-1600)/400) + 1
Now, let's say Bobby beats Bill. His new rating is as follows:
Rn = 1600 + (24 * (1 - 0.359935)) = 1615.36 = 1615
So after Bobby's first big win, he now has a rating of 1615. Now, let's see how Bill's rating changes due to the big loss:
1
We = ------------------------ = 0.640065
10^((1600-1700)/400) + 1
Rn = 1700 + (24 * (0 - 0.640065)) = 1684.65 = 1685
Given enough people and enough matches, the ratings eventually provide a rather accurate ranking of how good a player is in comparison to his or her competition. Often, ratings are used to rank the players in comparison to one another to determine tournament seedings and invitations.