Well, you get your group together and one of you goes to the mob who gives you the mission, at which point a window pops up and, depending on the mission you are doing, you all choose which kind of mob you will be inside the mission. Fairies, for example, everyone is the same. If you are doing the orc mission, you can be a warrior class, wiz class or cleric/mezzer depending. After you pick your class you zone into the mission (in the orcs and fairies one both use the same zone in place and same map within the mission). For some you gotta say something to the mob (pirate mission in BB for example) and he zones you into the mission. Once in the mission, you need to spend the AA points your monster has to spend and set up any appropriate hotkeys for those skills. At this point, if you like the way you have spent your points for that particular kind of monster, hit the save button in the AA screen and that will allow you to then hit the load button should you do that same mission again some time and want to quickly get the same AA's as you had this time. If you don't like the way you spent them this time, spend them different next time instead of hitting the load button.
Now you gotta actually do the mission. In fairies you are there to kill stuff...in orcs you are there to escort a drow out to safety....pirates you are collecting treasures as you sack and pillage Freeport. In the lavastorm mission one of you gets to play the roll of naggy the dragon and the others protect naggy from the hordes attacking.
Once the mission has been successfully accomplished, all of you zone out. In the fairy mission whoever looted the one of the early objectives has to be the first to zone out. In orcs, all zone out after the dark elf zones out. Essentially, always wait for the "You gain experience" message before you zone out.
Thats about it. Depending on the mission, there will be a certain time limit to accomplish the mission and if you get it done quicker than that, the remaining time left is the time you have to wait afterwards to be able to go on another mission.
I hope that helps you out.