In the graveyard scene in Goblet of Fire, when Voldemort's and Harry's wands connect, Priori Incantatem occurs (i.e. Voldemort's wand spits out the last spells it has performed). First, Wormtail's sacrificed hand emerges from the tip of Voldemort's wand, followed in order by Cedric Diggory, Frank Bryce the Riddle House caretaker, Bertha Jorkins, Lily Potter, and James Potter. They are able to converse in present time with Harry, and appear to have thoughts and desires - Cedric wants Harry to take his body back to his parents; Frank Bryce muses that Voldemort actually was a real wizard.
From Goblet of Fire - Chapter 34 - Priori Incantatem - British Hardcover edition:
At once, Voldemort’s wand began to emit echoing screams of pain ... then – Voldemort’s red eyes widened with shock – a dense, smoky hand flew out of the tip of it and vanished ... the ghost of the hand he had made Wormtail ... more shouts of pain ... and then something much larger began to blossom from Voldemort’s wand tip, a great, greyish something that looked as though it was made of the solidest, densest smoke ... it was a head ... now a chest and arms ... the torso of Cedric Diggory.
If ever Harry might have released his wand from shock, it would have been then, but instinct kept him clutching his wand tightly, so that the thread of golden light remained unbroken, even though the thick grey ghost of Cedric Diggory (was it a ghost? It looked so solid) emerged in its entirety from the end of Voldemort’s wand, as though it was squeezing itself out of a very narrow tunnel ... and this shade of Cedric stood up, and looked up and down the golden thread of light, and spoke.
‘Hold on, Harry,’ it said.
In Deathly Hallows when Harry opens the Snitch and accesses the Resurrection Stone, Lily, James, Sirius, and Lupin come to him, but it's made clear in the text that these incarnations of Harry's loved ones are a part of his mind and soul and are not ghosts (to Harry, they most resembled in form Tom Riddle from Tom Riddle's diary Horcrux) Also, it's evident throughout canon that neither James nor Lily remained as ghosts (neither feared death)
Harry had no personal connection to Wormtail's hand (aside from the fact it was used in Voldemort's resurrection potion), Bertha Jorkins, Frank Bryce, or even much of one with Cedric Diggory (they weren't chummy throughout Goblet of Fire, although polite to one another)
So what kind of specters does Priori Incantatum produce? Are they some kind of ghost? If not, what kind of spectral presence would they possibly be?
ETA: 01.07.12 The answer to this question is indeed in Goblet of Fire. I'm leaving this question open, though, for someone to answer if they find the information. Go check Goblet of Fire guys! :)