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Firenze is a centaur from the Forbidden Forest. He has agreed to help Professor Dumbledore in resisting Voldemort. But by choosing to join with Professor Dumbledore, a human, the rest of his herd have decided that Firenze is a traitor. The centaurs hate and resent humans – presumably in part because there are so many more of us, and we aren't so understanding of their needs – and poor Firenze gets caught in the crossfire. Hagrid interrupts the centaur herd trying to kill Firenze for going to help humans, so Firenze must now live up at the Hogwarts castle for his own protection. He cannot go near his home in the Forbidden Forest.
To give Firenze a reason to be at the castle, Professor Dumbledore appoints Firenze Divination instructor after Professor Umbridge fires Professor Trelawney. But centaur astrology is not at all like the tea leaves and crystal ball-gazing that Professor Trelawney has taught at Hogwarts in the past. Centaurs observe the larger movements of the stars, and they refuse to use these larger movements to predict "human nonsense" (27.43) like love or accidents.
Instead, the centaurs "watch the skies for great tides of evil or change that are sometimes marked there. It may take ten years to be sure of what we are seeing" (27.49). As Ron comments, Firenze is not very definite about anything: he tells his students to watch the stars, but to give up any hope that they'll ever be able to read anything accurately because they are (a) human, and thus kind of stupid, and (b) too young and untrained to know what they're looking at. Needless to say, this assessment does not please his students, especially Parvati Patil and Lavender Brown, who were both really taken with Professor Trelawney.
Talking to Firenze is the first real experience of a serious non-human perspective on the magical world that we get in the Harry Potter novels. We have heard a great deal from the house-elves, but they are bound to human households and have a vested interest in maintaining the human-elf status quo. The centaurs are completely different: they are serious, wise (mostly), and they reject human domination of the magical world. Firenze makes the wizarding world (with humans at the top and giants, centaurs, and house-elves lower down) seem less familiar, natural, and obvious. Why should wizards assume that they are so much better than the centaurs? Firenze seems pretty wise to us, after all.