A mission is the task of an agent or team of agents to spork a particular badfic. It is also the story written by a PPCer about a particular agent/team and a particular badfic; in this context missions are also known as PPC stories. Due to the prevalence of this type of PPC story, though, "mission" is also sometimes used generically to refer to any PPC story, whether there's a badfic involved or not. Context may be useful in determining which is meant.
Most missions are one-parters, operated on by a single team; however, due to badfics like "Subjugation" and "Rose Potter," there are also some multi-chaptered, multi-team operations. Most missions result in mad (well, more than they were before) agents and a mission report. Other typical occurrences in missions include creative language, one agent restraining the other, and at least one burnt-out CAD.
As alluded to above, not all PPC stories are missions. Some are the adventures of agents in non-action departments such as DoSAT or FicPsych; others can be agent backstories, interludes between missions, or speculative future or alternate universe stories.