What is the objective for League of Legends? Are there Quests?
Solution 1:
League of Legends is a MOBA, Multiplayer Online Battle Arena. You battle with your team to fight the enemy team to destroy towers guarding the enemy base, and then destroy said base. An average match of League of Legends lasts for about 20-50 minutes, though some matches may exceed this.
In the new playstyle, Dominion, your team tries to capture a majority of the 5 capture points on the map while defending the points you own. The team who can keep a majority of the points longest wins. These matches usually don't last more than 25 minutes.
There are (currently) 95 champions that any person can play, after unlocking them with real money or in-game points, each with several abilities and all with their own stats. By working with your team you can choose a character to support others, kill the enemy team, distract the enemy team or keep the enemy from acting.
Solution 2:
Forgot to mention 3v3, a game format very similar to 5v5 however it usually takes less time. And of course the number player is limited. There are no quests to say in the 5v5 but there are in dominion
Solution 3:
At the time this question was asked and answered, there was not a quest system. Now there is. Usually it just consists of "First Win" which gives an XP reward and resets daily.
Occasionally there are events which almost always come with some sort of quests as well. These can result in XP and in game money as well as shards for champions and skins, emotes, icons and more. Some of the events are PVE or mix but some are PVP only.
Recently (at least in North America), they also added a rewards system for watching the professional events (LCS in NA).
As noted in comments above, Dominion map has been removed. There are several rotating game maps to keep things interesting which stay for a limited period of time. They are announced slightly in advance and don't seem to have a regular pattern except (maybe ironically) around April Fools.
All of the above are 5v5 except for a very few PVE events which are 5 vs the map and one rotating event which was 3v3.
Not mentioned by anyone yet either is the ARAM game mode which is 5v5 PVP with a single lane where everyones champions are picked randomly from their pools rather than normally you can pick which champion you want.
Otherwise, much of the upvoted information remains the same today.