By: +David Herron; Date: April 23, 2017
The Three Doctors
Legendary renegade Time Lord, Omega, was causing serious havoc for Gallifrey. In response, the Time Lords sent the 2nd and 3rd Doctor's together, to be joined later by the 1st Doctor, to Omega's abode. Omega was described as the "solar engineer" who created the "supernova" from which the Time Lords drew the power required to run Gallifrey. He wanted to be free from his trap so Omega could travel the universe again.
When the 2nd Doctor appeared inside the TARDIS, he explained that the violation of the First Law of Time was deemed necessary by the seriousness of the situation. Hence, having multiple incarnations of a Time Lord in one place violates some kind of law.
Initially the Time Lords wanted to send only the 2nd and 3rd Doctor's, but their bickering drove the Time Lords to also send the 1st Doctor.
In other words - the theme of their relationship was hostility towards each other.
The Five Doctors
Another legendary renegade Time Lord, Rassilon, had been trapped in a crypt in the Death Zone on Gallifrey.
Lord President Borusa arranged to use a "Time Scoop" to bring five incarnations of The Doctor, various of his Companions, plus The Master, into the Death Zone. He was playing an insane power game having learned that Rassilon could offer immortality. That turned out to be a trap, however.
But what we're interested in is their interaction. In this case there was a bit of bickering, but generally they got along.
The Two Doctors
The 2nd Doctor and Jaime were sent by the Time Lords to contact space/time researchers on a remote space station. The Time Lords wished to warn of damage to the fabric of time by the research. However, one of the staff has some other plans involving stealing the technology for their own purpose.
That leads to a situation where the 2nd Doctor, Jaime, the 6th Doctor, and Peri, are collaborating to stop the risk.
The two Doctor's were separated for most of the story, but once they met up they tended to get along.
Technically this isn't a multi-Doctor adventure, since it involves a created version of the Doctor rather than multiple incarnations. In this story the Daleks had concocted a crazy scheme involving stealing 27 planets from around the cosmos for some kind of crazy energy something or other that would create a Reality Bomb that would destroy all universes everywhere. Why? Who knows. But, the story involved two instances of the 10th Doctor.
The 10th Doctor had begun a regeneration after being shot by a Dalek. But, since David Tennants contract with the BBC hadn't finished, he shot the Doctor regeneration energy into his spare hand that was stored in a tank in the TARDIS console room. Later, that spare hand regenerated into an instance of the Doctor which was imbued with human whatever due to the presence of Donna Noble.
Eventually the human-Doctor committed a genocide, because he destroyed the Daleks and their Crucible. For this "sin" he was exiled to the alternate dimension in which Rose Tyler lived. As a human-Doctor, he had only one heart, could not regenerate, and would grow old and die in a normal human rate so that he and Rose could live happily ever after together.
The Almost People
Technically this doesn't count as a multi-Doctor adventure, since it involves a simulated version of the Doctor rather than multiple incarnations. In this story, The Doctor and Amy and Rory landed on an island where simulated people were used to conduct dangerous life-threatening people. Simulated as in, bodies were grown in a tank of goop and that the "real" person would animate the simulated body. The simulated body could do a dangerous thing, and be killed, and the "real" person would be able to live. Sometime later the technology was seemingly used to create a fake version of Amy Pond, as well as a fake version of Melody Pond as a baby.
Anyway, this setup gave the opportunity for there to be two versions of the 11th Doctor. This involved some creepy narcissist self love thing.
The Day of the Doctor
For the 50th Anniversary, on November 23, 2013, the BBC set up a multi-Doctor story involving the 10th and 11th incarnations, as well as a previously unknown incarnation, The War Doctor, occurring between the 8th Doctor (Paul McGann) and the 9th Doctor (Christopher Eccleston). The major issue in the story was The Time War and a rewriting of that history and the Doctor's involvement in its ending.
While most of the action involved those three Doctors working together (of sorts), the culmination involved all 12 (no, 13) incarnations of The Doctor. The idea presented in the story is that a psychic message/idea was sent to all his incarnations, so that all 13 instances of the TARDIS could be employed to pop Gallifrey out of its time stream.
As for the relationship, there is a bit of friction between the Doctor's but not a lot.