Lanes are the main paths that minions follow in the course of a game. The Lanes are defended by turrets, and in order to push the enemy into their base and destroy their inhibitors and their Nexus, each successive turret in a lane must be destroyed before damage can be done to the next. Along each lane is an abundance of brush for tactical maneuvering such as juking and ganking, and controlling a lane is a key tactical component of gameplay; early lane control in a game can put your opponents at a disadvantage by keeping them away from your turrets and denying last hits on minions for much-needed gold.
Summoner's Rift has 3 lanes (Top, Bottom and Middle) which are defended by 3 turrets (Outer, Inner, Inhibitor). Each adjacent lane is connected from different points through the Jungle or by the river that runs diagonally through the map.
In lower-level games, the general rule is that one champion goes top, one goes to the jungle, two go bottom, and one goes in the middle. If you are having trouble with the champion(s) you are facing, you should ask someone to switch with you.
Over level 30 it starts to get more complicated. There are many combinations, but the most popular metagame at the moment tends to follow a set pattern of one jungler with a tanky-DPS at top, an ability power reliant caster will go middle and the carry going bottom along with a support champion.
The Twisted Treeline has two lanes, top and bottom, that are defended by 2 Turrets (Outer, Inhibitor). Typically, the top lane is the solo lane, with the other two team members trying to gain a foothold in the bottom lane. The narrower distance between lanes demands more care against sudden ganks. Top Lane takes a longer time to get from the first turret to the second turret while the bottom lane is shorter from the first turret to the second turret.