The Elo rating system is a method for calculating the relative skill levels of players in two-player games such as chess. It is named after its creator  The Elo system was invented as an improved chess rating system, but today it has been adapted for use in many other games. Variations of it is also used as a rating system for multiplayer competition in a number of games and has been adapted to team sports including association football, American college football and basketball, and Major League Baseball., a Hungarian physics professor and chessplayer.
In League of Legends the Elo rating of a player is used by the matchmaking in ranked games to find other players of a similar skill level to play with/against. Elo is not used for custom and Co-op vs. AI games. The Elo rating for ranked games is different for each queue types: 3v3 arranged, 5v5 solo and 5v5 arranged teams. The rating is only visible for ranked games after 10 games played in a certain queue type. A summoner's Normal game Elo remains hidden at all times and can only be guessed upon based off his or her win/loss ratio and the apparent skill of teammates and enemies, although not perfectly accurate as normal queues are less stringent as opposed to ranked queues. Players were awarded with medals in their summoner profile based on their Elo at the conclusion of Season One. These medals are given as follows:
- Bronze: Between 1250 and 1399 (3v3: 1249-1409, pre-made 5v5: 1249-1409) (Top 25%)
- Silver: Between 1400 and 1519 (3v3: 1410-1519, pre-made 5v5: 1410-1499) (Top 10%)
- Gold: Between 1520 and 1899 (3v3: 1520-1699, pre-made 5v5: 1500-1749) (Top 3%)
- Platinum: 1900 and above (3v3: 1700+, pre-made 5v5: 1750+) (Top 0.2%)
About a month before the end of Season 2, a new rating tiers system was introduced:
- Bronze: Between 0 and 1149 (Team: 0-1249) (Top 100%)
- Silver: Between 1150 and 1499(Team: 1250-1449) (Top 40-50%)
- Gold: Between 1500 and 1849 (Team: 1450-1649) (Top 5-10%)
- Platinum: Between 1850 and 2199 (Team: 1650-1849) (Top 0.5-1%)
- Diamond: 2200 and above (Team: 1850+) (Top 0.05-0.1%)
In addition, rankings are subdivided into smaller sections, like Bronze VII or Gold I. These are sectioned off every fifty points of elo (1150-1199 is Silver VII, 1200-1249 is Silver VI, etc.).
These are the tiers for the North American platform, they differ a bit on other platforms.
The math of Elo
The specific formulas used for Elo calculations in League of Legends are unknown. However, most Elo implementations share the same basics as that originally designed for chess. A brief summary is given below. For a more detailed discussion, see.
It is assumed that a person's performance varies from game to game in approximately aand a person's Elo rating is the mean of that distribution. A person with a higher Elo will perform better on average than a player with a lower Elo. This score is determined entirely by win/loss statistics in relation to other players. For players A and B with respective Elo ratings of Ra and Rb the expected victorious outcome Ea of the game for player A is given by the following formula:
For every difference of 400 points, the team/player with the higher score is ten times as likely to win as the other team/player. This standard is for Chess and may be different in League of Legends. After a game the actual outcome is compared to the expected outcome and each team/players rating is adjusted to bring them closer to where they should actually be. As a result, if a team was expected to win and does their score changes less than if they where expected to lose and instead won. Successive games should eventually bring each player/team to a point where they are expected to win 50% of the time against opponents of equal score.
A player's change in rating is linear to the difference between the expected outcome and the actual outcome. It is given by the following formula.
The magnitude of the score change is determined by the player's K value and other things, like a server crash (When the score you get is 50%). In chess initially this K value is big (25 for their first 30 games) resulting in large changes in Elo. This is so a player can rapidly find his correct place in the ranking system. As their number of wins and losses becomes more even this K value is reduced to prevent dramatic changes in Elo against evenly matched opponents (K = 15 to 7). This also prevents inflation in ratings at high Elo play. It appears that League of Legends uses a similar system of changing K values: K appears to start around 60, eventually leveling out to about 25.
All players start ranked play with an Elo of 1200 for their first 10 games at level 30. From there they are assigned a score and changes are made as normal. 
Elo decays over time when you are above 1400 Elo:
- Elo decays at a rate of 25 Elo for every 4 consecutive weeks of inactivity.
- For normal rating, inactivity is defined as no activity in any queue.
- For ranked rating, inactivity is defined as no activity in the specific queue (arranged 5x5, arranged 3x3, and solo/duo 5x5 are all tracked separately). Ranked decay only applies to people who are ranked at or above 1400 rating.
- The decay timer is reset after a game is played in that specific queue.
- Elo will not decay below a rating of 1400.