Tenebrus is a Thestral. The Thestrals are skeletal, winged horses that are attracted to the smell of death. They eat raw meat. They can only be seen by people who have witnessed death, of whom Harry is now one. Apparently, the "horseless" carriages that serve Hogwarts have been pulled by Thestrals all of this time; Harry just hadn't seen enough death up close and in person to recognize these animals.
Harry's ability to see them now demonstrates exactly how much Cedric's death has changed him. We now have an objective measure of the power death has over people: those who have watched it can see Thestrals; those who haven't can't. When Ron rides his Thestral to the Ministry of Magic at the end of the book, he can't get over how strange it is to ride through the sky on a flying horse he can't see. Frankly, we can't imagine it at all. All the same, the Thestral gets them to the Ministry of Magic when they want to go, which is the most important thing.
Tenebrus is Hagrid's "special favorite" (21.56) Thestral. He was the first Thestral to be born in the forest.