Players are formed into 2 even teams with 1-5 champions on each team. Each team starts at opposing sides of a map, near what is called a "Nexus". A match is won when either the opposing team's Nexus is destroyed or the other team surrenders. To destroy a Nexus, each team must work through a series of towers called "Turrets". Turrets are often placed along a path to each base referred to as a "Lane". Along the way, each player must gain levels from killing the opposing team's "Champions" and "Minions" (small NPCs that constantly spawn and attack the other team) and defeating neutral monsters (some of which grant buffs known as Crests upon death). Also, champions purchase "Items" with in-match gold to improve their champion. In League of Legends, each player starts at level 1 at the beginning of the match and can obtain the maximum of level 18 with their champion, leveling 4 different champion-specific abilities.
Source: Wikipedia, "League of Legends", available under the CC-BY-SA License.