It's the Elo Rating System, originally developed in chess to determine your skill level relative to other players. Basically, if you win a game you gain Elo depending on how much "better" their team was compared to your team. To use a simple system:

To explain the theory behind it, say your team is 5 people with 1300 Elo. Their team is 5 people with 1200 Elo. A naive method (but not an unreasonable one) would say hey - you're supposed to win this match. Normally, you would gain, say, 8 Elo for winning and lose 8 Elo for losing. But because your team is favored to win (higher average Elo), these effects are skewed: you would gain, say, 6 Elo for winning or lose 10 for losing. The other team would gain 10 Elo if they win or lose 6 if they lose. Over the course of many games, you find an Elo that approximates your skill level.

Your Elo can be checked in Ranked games on your summoner profile. Outside of Ranked, it is hidden (but it does exist for Normal games).

EDIT: See clarification below.

A side note, while the Elo system was developed for chess, LoL uses a heavily modified version that accounts for the differences in multiplayer. When people talk about 'Elo systems,' it's a category with the same general means of measuring skill (compared to say, win % or win count).

For normal matches...