In League of Legends, you will start at level 1 when you create a new account and the max level is 30.
Every level will give you a point in Mastery, open up new summoner spells and once you reach level 30, you will be able to play on ranked game (more on ranked game later)
So here is the mastery screen and it showing my solo top Teemo page (taken from solomid.net by TheRainMan)
As you can see there are 3 different mastery trees (Offensive, Defensive and Support), all mastery will affect your in-game stats.
When you gain level you will also gain extra summoner spells. Every player will be able to pick 2 summoner spells to be used in game, majority of players will pick Flash, because of its function.