Quests

Quests are things you find out about and might want to try to do. They contain and form a lot of the story of GECCTT. Or you might completely ignore them and go on about your business of annihilating other gangs. However quests usually have a reward. More XP, more technology or more items and not the least possibly powerful friends. It can be worth it having a mega-corporation on your side. So they are usually worth attempting.
If anyone else has a quest it can be worthwhile using that distraction to kill them or let them nearly finish it and steal the prize after they soften it up for you.

Some quests are just announced as text on screen for all to see. eg one gang may have a quest to kill Lord Polyp and as soon as one of his guards is killed the message appears on all screens "I will pay 30mins of cash to the gang that gets me safely to the WWC base."
Others are given to specific gangs due to their political affiliations and show during the loading screen.
Others are never announced. You need to figure out that they are possible from clues you have earlier gathered.
The trick with these quests is often to know they are happening. You might have read about the temple of Goterdamn when you bought your plasma rifle but you've never seen the thing. Then a few missions later you notice the rune described on a wall. Time to get to work, you've got a long lost relic to find.

One thing with quests is you often have a choice as to how to go about them. Do you want to kill the leader of the faction and be rewarded by his enemies or warn him? Each quest and the path you take has a consequence on future quests and how even some wandering monsters treat you. So your choice may depend on who you want to be friends with.

Some common quests are:

Kill em all!
Destroy all of a special monster type. May require some knowledge. eg the final monster lives in a cave that can only be openned by a plasma weapon. They might only be able to be killed in a Coup De Grace.
Rescue
Find someone and get them away from where they are. Or find them and stop the assasination plot, usually still requires taking them somewhere (without being shot)
Assasinate
Kill some specific enemy. might be guarded. Fun if another gang decides to rescue them.
Hold the location
Hold a specific location. Perhaps have to stop the enemy from destroying it.
Find the thingy
Find some special item. Steal it, blow it up or take it somewhere.
A document, a computer etc.
Find the way in
Many missions will include this. to complete it you must find and enter a special territory. Eg first go to the blue tower, this will open a switch at the red tower, which when flicked will open the temple of gotterdamn. General Quake FPS type mission.
Some servers may run Adventures and Campaigns. Adding greater depth to the game rather than the randomly generated quests on most servers.

Adventures are a bunch of quests linked.
eg find the entrance to the research lab, kidnap the scientist, take them to the holding area until the car arrives. All while another gang has been given the job by Amalgamated Meditech to keep their researcher safe and yet another gang has been hired to assasinate him and keep him out of both sides hands.

Campaigns will be linked Adventures with server side saved values. After finishing one adventure (a bunch of quests on a certain map) a Campaign may set the next map and the map objects and the results of the last Adventures Quests affect the next adventure in the campaign.
eg the scientist was delivered, but now you find the company that hired you has decided it's safer to kill you to keep you quiet and not ruin their standing with AMI. The assasins haven't given up and might not have been told your present standing with your ex-employers. And you're not even sure who that was, but you want to know...