In both Elyos and Asmodian main cities there is a Coliseum. Going here you will find a large open area, much like the ones the Romans used to fight in. Speak to the NPC out front in order to enter.

The coliseum is an open free-for-all battle place where you are able to test out your skills and/or strategies against other players of the same team (if you are Elyos you can only fight Elyos, if you are Asmodian you can only fight Asmodian) with no experience loss on death.

When you enter, you can go either alone, in a group, or in an alliance. When you are with a group or alliance you will be unable to attack your allies.

All of your enemies will have red names, easily separating them from yourself or your allies. This makes setting up organized battles extremely easy (although randoms can and sometimes will still teleport in and start jumping in the middle of battles).

Upon death, you will automatically be teleported back out of the Coliseum and you, will be placed right next to the entrance NPC. You can then either rest back your healt and mana or you can enter in again!

The possibilities for using the Coliseum are endless, and going there as a Legion could help build class strengths and learn how to properly PvP against others.
 
Quests
When it comes to questing in Aion you will run in to three different types: Work Orders, Missions, and Quests. The only quests that are required to progress your character are the ones listed under Missions. Anything you find under Quests or Work Orders are completely voluntary, although they do (often) yield good rewards.

Missions/Campaigns
Missions are required to boost your character, so they should be the top priority for you in terms of progression. For example, at the beginning of the game you will get a Mission that has multiple quest parts and requires you to be level 9 for the final one.

Note that completing all of the various missions will yield great rewards, and they will significantly help you level up your character. The level 10-20 ones, for example, will easily gain you at least a few levels while working on them!

Quests
These include all of the normal quests you take part in. These are also important just because they will lower the amount of mob grinding you must deal with later. Also, these can give great benefits like Stigma Stones, weapons, armor, recipes, etc.

Quests are not required but are highly recommended to do. If you are working on one and decide it is not worth your time (either you are having issues figuring it out or you have already out-leveled it) you are able to drop it. If you drop a quest on accident you are able to get it again by speaking to the same NPC you picked it up from before.

Work Orders
Work Orders are a very unique type of quests, as they are required to boost your profession level. If you are doing weaponsmithing, for example, you will need to do quests so you are able to do better and better weapons. These quests you are taking will be listed under the Work Orders section of your quest log.