League of Legends is a free game, which means that you have to download the client before playing. Before you can download the client or play the game, you need to first create an account on the League of Legends website. After you launch the client for the first time and log into your account, you will then choose your "Summoner Name" (the name that gets displayed to others when playing games).
The premise of League of Legends is that you control one particular champion on a team of 2 to 4 other people, where you must work together as a team to destroy the opponent's nexus in the center of their base. To get to their nexus, you need to work your way through one or more of the "lanes" on the map, destroying enemy towers on the way.
In addition to the gameplay itself, League of Legends has a metagame system where you can earn Experience Points (XP) and Influence Points (IP) through playing games. Wins grant more XP and IP than losses, although playing a game through to its completion guarantees that you will earn at least some XP and IP (except in certain situations with custom games, such as imbalanced teams). The XP allows you to level up out of game, which unlocks mastery points and rune slots. The IP earned allows you purchase new champions or runes in their store.