The Tribunal originally appeared in a single unexplained issue of the Defenders, written by Ed Hannigan, as a seemingly rogue government agency conducting an inquiry into the Defenders.
Years later, J.M. DeMatteis tied the Tribunal and the equally-never-explained Elf with a Gun into a grand retcon that gave both a complex backstory.
Years after that, in Peter David's Incredible Hulk vol. 2 #370, Dr. Strange claimed that DeMatteis's story was all a cosmic hoax but did not elaborate on the subject. The current state of the truth about the Tribunal is unknown.
Major Story Arcs
The Tribunal is a mysterious cosmic hive mind, claiming to be the antibodies of the Universe. They reportedly dwell outside time itself, locate disease in the cosmic body, and work to eradicate it.
At some point in the future, they discover that the Earth had been destroyed before its inhabitants had reached their full potential. They decide to see if they can stop this from happening, but first they have to find out why it happened.
They create their Time Agents, often known as the Elves with a Gun, and send them to various points in humanity's history. They kidnap some humans who seem to be involved in the catastrophe, but it has no effect. They eventually determine that the Defenders are involved, and hold an inquest, inviting various friends and enemies of the Defenders to come and relate their knowledge of the team. This isn't enough, so they create a Time Buoy robot to spy on them. This robot, known as Luann Bloom, malfunctions and starts to think it's human, so it has to be recalled by the Time Agents.
They ultimately discover that the events are connected to the original Defenders, Dr. Strange, Hulk, Namor and Silver Surfer, acting together. They have a Time Agent bring them to the future and explain the situation: the Defenders will eventually come upon an alien ship, and the aliens on board will kill themselves thinking that they have been discovered by their home empire, which seeks revenge on them. The alien empire does eventually find out about the crashed ship, and destroys the entire Earth in order to eliminate their bodies, in accordance with their customs. The only way to stop this is for the original Defenders to break up. The four agreed to never team-up again to prevent the events coming to place.
However, Dr. Strange looks into the whole thing later on, and finds out that it was a cosmic hoax of some kind. Since the nature of this hoax isn't explained, it is unknown what exactly was true or untrue about the story. Perhaps it was only the story of the alien ship that was untrue. Perhaps the Tribunal or the Time Agents were not even who they said they were. However, when the Tribunal first brought the Defenders to its time to talk with them, Strange looked at the Tribunal with the Eye of Agamotto, and said that the Tribunal was in fact who it said it was (this is backed up by the fact that the Surfer said he had heard of the Tribunal before, and that even Galactus spoke of them in whispers). This implies that perhaps only the story of the alien ship was false. However, an Elf with a Gun has shown up in recent years, and showed no relation to the Time Agents or the Tribunal. This implies that perhaps the Time Agents were not necessarily who the Tribunal said they were either, or at the very least that the Time Agents are not the same as the original Elf with a Gun.
Powers and Abilities
The Tribunal is a powerful cosmic entity. They are omnipresent in time, but are unable to incarnate themselves in a particular past time period without causing great damage to that period.
They have great scientific knowledge, having created time travel technology, Time Buoy robots, and the Time Agents, who seem to be organic clones.
Some of their more minor shows of powers include putting someone to sleep, causing great pain in Namor, creating illusions, and telepathy.
They were also apparently capable of temporarily eliminating and then recreating powerful beings such as the Surfer and Dr. Strange without them being able to do anything about it.
Despite their great power, they are capable of error when they take on humanoid form.