Assassination/PTQ - Kill a single unit/Protect a single unit
Bound - Navigate a unit through a lethal obstacle course
Defense/Offense - Protect an area/Assault an area
Diplomacy - Effectively manage alliances and controlled areas to destroy all opposition
Madness - Use units that continually spawn at a preset location to eliminate other players
Melee - Played like regular melee StarCraft with some type of variation
Movie - Sit back and watch
RP - Use unit spawning tools to create your own environment, then roleplay with the other players in it
RPG - Pick a unit and follow the objectives given to you through the storyline
That's some of the major ones, although there are others, including many miscellaneous ones which you'll simply have to read the briefing to know the objectives and gameplay of.