In common fan parlance, a multi-Doctor story is one that involves more than one incarnation of the Doctor. Typically, although not necessarily, such tales are reserved for anniversary and/or charity events. The term is not usually applied to stories like The Almost People which involves a doppelgänger, or Father's Day, which has multiple instances of a single incarnation.
They are exceedingly rare on television and audio, but much more common in short stories and comics.
Multi-Doctor stories include:
Many aspects of the multi-Doctor story have been explored over-the-years in many different forms. The Multi-Doctor story bases itself in the bending of the Laws of Time, as the Doctor is explicitly disallowed form crossing his own time-stream.
A contentious topics for fans (and for a time, only fans) was if the characters featured in Multi-Doctor stories would actually be allowed to remember the events of the story afterwards. If, for instance, the Second Doctor did not forget the events of TV: The Three Doctors, then it would make little sense for the Third Doctor to not also remember the same events. Many fans speculated that this meant that the Time Lords simply wiped the Second and First Doctors' memories of the story. This some somewhat supported by such stories as PROSE: The Empire of Glass and PROSE: Briefly Noted.
Writers at the time, however, were not concerned with this continuity point. Indeed, in the televised story TV: The Five Doctors the first, second, and third incarnations of the Doctor speak as if they remember their first encounter when meeting in Rassilon's tomb for the first time.
An early script for the TV: Dimensions in Time featured a scene where Jamie McCrimmon would suddenly find himself next to the Sixth Doctor, leading Jamie to recognise him from their encounter in TV: The Two Doctors. This scene was cut after Frazer Hines realised that he would not be able to meet the schedule of production.
One of the most notable examples of versions of the Doctor remembering the events of the story was in 2007's Time Crash, where a central plot point to the Fifth and Tenth Doctor's survival in the story is that the Tenth Doctor remembers seeing himself save the day from the point-of-view of the Fifth Doctor.
Since these stories, most mainstream and secondary stories have to come to accept the idea that characters lose their memories after the events of Multi-Doctor stories. TV: The Day of the Doctor features the realisation that the Tenth Doctor and the War incarnation of the Doctor will both forget the events of the story in due time. COMIC: Four Doctors expanded on this, stating that in multi-Doctor events, all but the final incarnation of the Doctor will forget about the story's taking place. His companions, however, will remember. Later stories contradicted this, with COMIC: Lady of the Blue Box suggesting that Gabby barely remembers the events of Four Doctor and COMIC: The Heralds of Destruction suggesting that even the Third Doctor had all-but forgotten the events of The Three Doctors.
Like most things in the impossibly large spectrum of Doctor Who content, deciding which of these accounts hold true is up to each and every fan to decide.