Brannigan was one of Red Dwarf's psychiatrists.
He was an excitable, upper-class man, whose personality had been uploaded into the holographic database. Brannigan was presumably killed in the accident that wiped out the crew of Red Dwarf.
When a meteor struck the ship three million years later and damaged the Hologram Simulation Suite, Rimmer began to mimic the personalities of various crewmembers. The first of these was Brannigan, who hopped around and explained that he was late because of the shuttle, which was usually brilliant. Lister recognised Brannigan's personality, leading Holly to realise that the damage was very serious. ("Queeg")
Behind the scenes[]
The scenes where Rimmer took on the personalities of Brannigan and the others were written to showcase the impersonation skills of Chris Barrie.
Since Lucas McClaren later described himself as the ship's "Chief psychiatric counsellor", it is possible that Brannigan was a deputy or had left Red Dwarf before the accident. An interview between McClaren and Talkie Toaster on the official Red Dwarf website reveals that the latter is the case, and that Brannigan was McClaren's predecessor.