Hippogriffs are flying creatures with the head, wings, and forelegs of a giant eagle and the body (including hind legs and tail) of a horse. Their eyes are orange, while individual hippogriff colors vary as those of mundane horses do, including black, bronze, chestnut, grey, and roan. An adult hippogriff’s wingspan is approximately 24 feet.
Hippogriffs are carnivorous and are extremely dangerous until tamed, which should only be attempted by a trained witch or wizard. That said, hippogriffs can and do live on insects, birds and small animals such as rats and ferrets.
A person wishing to approach a hippogriff should maintain eye contact and should bow first; if the animal bows in return, it can be touched and even ridden. Hippogriff owners are required to keep them under Disillusionment Charms to prevent Muggles from seeing them.
Hogwarts has access to at least a dozen hippogriffs for Care of Magical Creatures lessons; see Buckbeak for an example.
From the Greek hippos (horse) and the Late Latin gryphus (griffin) (Online Etymology Dictionary)
The hippogriff is a legendary creature born of the union of a horse and a gryphon. Since horses are the natural prey of gryphons, the fact that they would mate makes the hippogriff a symbol of love and impossibility.
Rowling discussing with Alfonso Cuaron how she created her version of the hppogriff:
Jo Rowling: I think it's important to say I didn't invent the hippogriff. I invented that hippogriff, but the creature the hippogriff, as you know, is in folklore and mythology, so that's not my creation. But I really thought hard about this, because it could've been, in the book, it could've ben an absurdity. And indeed, it really could've been in the film as well, but I thought you made him a real creature.
Alfonso Cuaron: There are not that many graphical representations of hippogriff, and that is something with the story that is very interesting. There are sphinx, there are several sphinx, or you see creatures that are half bird and half cat, a lot of different things. But for hippogriff, it was actually hard to find....
Jo Rowling: I knew that 'cause I went looking.
Alfonso Cuaron: You knew that, yeah.
Jo Rowling: I could hardly find any anywhere.
Alfonso Cuaron: No, I know.
Jo Rowling: So I thought it's complete liberty to invent. (PA/dvd)