An Ambassador-class Federation starship commanded by Captain Darson, whose first contact mission to Ghorusda ended in the deaths of forty-seven people including the captain.
The USS Adelphi was the ship involved in what Starfleet records call the Ghorusda disaster, described in the third-season Next Generation episode Tin Man.
The Adelphi carried a first contact team to the Ghorusdans with the Betazoid specialist Tam Elbrun as adviser. Something went badly wrong, and forty-seven people died, among them Captain Darson and two of William Riker's Academy classmates. The official inquiry cleared Elbrun and laid the blame on Darson's judgement, but many officers, Riker among them, went on holding Elbrun responsible.
That history explains the cold reception Elbrun gets when he comes aboard the Enterprise-D for the mission to the living ship Gomtuu. The Adelphi itself is not seen.