Each time you level up you unlock a slot to place a rune. Runes can be bought from the market on the client using the IP that you get for playing games. Each rune you equip will slightly boost a stat for your characters depending on which runes you buy. Tier 1 runes are available from the start but aren't hugely important. Upon reaching level 20 you can start buying tier 3 runes that are much more noticeable. Keep in mind that these can not be bought with RP (which you get by spending money). --- http://steamcommunity.com/id/LETITRIIIIIIIIIIIIIIIP
Each level for your account lets you use 1 more rune. So a level 30 account can have 30 of 'em, while at level 3 you only can use 3.
Runes themselves are things you can buy in the league shop (only with the in-game IP points, not with real money). They give stat bonuses that are carried over into your character when you start the game, kind of like mastery pages if you've taken a look at those yet.
One of the main reasons you'd want runes is for jungling, as was previously mentioned. Jungle monsters hurt more than lane minions, so if you're gonna jungle, you'd want to set up a rune page that lets you start the game with extra armor, along with more damage or attack speed to hurt them faster too.
Knowing which sort of runes to buy is something that takes some time and experience with the game to understand, and they're as relevant until your account is closer to level 30. But it doesn't hurt to learn a bit about 'em now. --- And then John was a zombie.