Acromantulas
Acromantulas named Aragog and Mosag, together with their large family, lived in a dome-shaped nest in the very heart of the forest until Aragog's death during Harry's sixth year (HBP22). Hundreds of the giant spiders now dwell there, all decended from the original pair. Aragog was a pet of Hagrid's fifty years ago (CS15). The Acromantulas are among the most dangerous creatures in the forest (FB), although Grawp may have taken the title of 'most dangerous' away from them now.

centaurs
At least 50 centaurs of fighting age live in the forest (OP33). In typical centaur fashion, most of them avoid humans and do not involve themselves in human activities, simply watching portents in the stars and uttering cryptic responses to questions.

(Young Firenze is an exception, and was cast out from the herd and nearly killed because of it in Harry's fifth year.)

Hagrid knows many of them by name and had a civil (though sometimes frustrating) working relationship with them until partway through Harry's fifth year. (Even in Harry's first year, he said that the centaurs weren't exactly friendly.) When Hagrid brought Grawp home with him late in 1995 [Y15], the centaurs objected, and when Firenze accepted a job offer from Dumbledore, the centaurs turned against the staff and students of Hogwarts castle. See centaurs for more details about the Forbidden Forest herd (PS15, OP30).

Fluffy
When he was no longer needed to guard the trap door on the third floor, Fluffy the three-headed dog was released into the forest [2].

a wild turquoise Ford Anglia
Here we have something that is not a living creature, but perhaps worthy of mention. The Ford Anglia was an automobile owned and enchanted by Arthur Weasley. After a bad fright and an exhausting trip across the length of Britain, the Ford Anglia escaped into the forest. There it became wild and took to trundling around among the trees (CS15).

Grawp the giant
When Hagrid and Madame Maxime acted as envoys to the giants in the summer and autumn of 1995 [Y15], Hagrid learned of his mother's death and of the existence of a half-brother he'd never known: Grawp, a full-blooded giant. Since Hagrid has no other family, and Grawp (being a runt for a giant at only 16 feet) was being bullied, Hagrid managed to smuggle Grawp all the way back to the forest. This was the last straw for some of the forest's inhabitants, specifically the centaurs (OP30).

thestrals
A herd of about 100 thestrals lives in the forest; Hagrid says proudly that he's probably the only person in Britain to have domesticated them (OP21). Grubbly-Plank says he's got them well-trained (OP17). Hagrid waited until Harry's fifth year to cover them in Care of Magical Creatures, considering them a special treat.

unicorns
The unicorns are powerfully magical creatures living in the forest. They are extremely fast and are seldom hurt. Hagrid found one dead in the spring of 1992 [Y12] and signs that another had been horribly wounded, and spent a night searching the forest for both the injured unicorn and whatever could have done such a terrible thing (PS15). In the ordinary way, the unicorns shed hair that becomes tangled in the undergrowth of the forest; Hagrid collects it to use as bindings on bandages for various injured animals because of its strength (HBP22).

The following creatures are alleged to live in the forbidden forest, but not on reliable testimony:

trolls
Many years ago, when Hagrid was a boy and a student at Hogwarts, he got in trouble a few times with the headmaster, Armando Dippet; Tom Riddle claimed that one reason for this was for sneaking off into the forest to wrestle trolls (CS17). We only have Tom Riddle's word for this, and he slandered Hagrid about "werewolf cubs" in the same sentence, so the alleged presence of trolls in the forest may merely be another flight of fancy on Riddle's part.)

werewolves
Draco Malfoy said before his first year detention that he'd heard werewolves live in the forest, but Hagrid didn't affirm or deny this. When Harry subsequently asked whether a werewolf could have been killing the unicorns, Hagrid only answered the specific question asked, saying that a werewolf isn't fast enough to do that (PS15). (And after all, even if no werewolves live in the forest, a werewolf could have been getting in from the outside from time to time.)