Been wanting to play but I don't know if its worth playing. What is the objective of the game?
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.
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.
There are many different game modes, Dominion, 5 V 5, 3 v 3, Co-op, Custom Games. The objective changes based on the game style you want to play however a couple of them are similar with some small difference.
Dominion - Capture Points and prevent the other team from capturing points. It's basically a circle and you control you champion to reach certain points. Your team starts at 500 life as well as the other team. Killing enemy champions and capping points reduce the health of the other team.
5v5, 3v3 - this is played against other players, you are playing a chosen champion with 4/2 other teams and their champions. You move down lanes and take out defending towers so that you can move on to the enemy base like area and destroy their main "life" known as the Nexus. Doing this all the while the enemy is doing the same to you. So think of it like playing capture the base as a kid with laser tag guns. Make your way to the fort, taking out all that you can. If you're hit, you go back to base and revive.
Co-Op - is a bit decieving, This is based off of 5v5 and dominion. You play with 4 other real players however, you do face AI/bots instead of other human players. Same concept as the other games but just a little less competitive. You can say it's a learning arena
Custom Game - same as all the other game types just you are more so allowed to establish the rules/game type and who plays.
The objective is to have fun :)
LoL is a purely match-based PvP game (there are bots, but they're none too smart), so each match pretty much stands on its own. Riot does have a fairly comprehensive lore attached to the world, and each character, but the player doesn't experience any kind of story. The game isn't any kind of RPG, just an endless supply of deathmatches :)
You do have a single meta-character (called a summoner) which gains XP with each game, and there's a level cap of 30. However, each level only grants a fairly small increase to character power (in comparison to most RPGs, for instance) and it doesn't take much playing to get to the maximum level.
The characters you actually play each game with have a fixed skill-set, and they level up, gain gold items and money entirely within each match, and are reset at the start of each match. You can play with any of the 'free' characters (a small selection of characters which changes each week), or characters you've bought with in-game currency or real money. You can also buy runes (which are the closest thing LoL has to gear) with in-game currency, and attach them to your summoner. They confer small bonuses which apply to whatever character you select when actually playing a game.
For most people, the main reason for playing is to earn more in-game currency which will let you buy new characters. Each character has a unique skillset and playstyle, and although most players will develop an affinity for a particular character, its nice to be able to pick from a stable of characters if you feel like mixing it up.