1. Were it not for Helga Hufflepuff, the centaurs of the Forbidden Forest would never have settled there. They were originally running from the wild human tribes, and were pushed into the Forest by a particularly rowdy force. The Founders helped the centaurs fight off the Muggle force, but the whole ordeal prompted a summit between the Founders and the centaurs. There was a murmur of banishing the centaurs; however, Helga stuck up for them, proclaiming that they had as much right to be here as the wizards did. Thus, the centaurs were allowed to stay, and a sanction on their behalf was eventually signed.
2. Centaurs are an all-male species; however, their subtle magics allow them to breed with females of multiple species. In the old days, they stole away maidens for this purpose; nowadays, they have to get a breeding permit and enter an agreement with their maiden of choice. Said permits are essentially the only legal activity the Centaur Liaison Office sees.
3. Their homeland will always be Greece, but after the fall of the Roman Empire, centaurs at large were driven from that home. Nowadays, they're scattered all around the Mediterranean and beyond, from as west as Ireland, to as south as Afghanistan, to as north as Finland, and to as east as Mongolia. The Hogwarts herd is one out of 5 in Britain (which, incidentally, has the most individual herds of any country).
4. Up until 1803, centaurs were an active part of the wizarding community, and herds sometimes passed through towns and villages. However, that year marked the Ban on Commercial Anthropophagy, which stated that no carnivorous being would legally be able to consume authentic human bodily products (blood, flesh, etc.). Thus, vampires, hags, and the like turned on the other beings of the wizarding world, especially centaurs and mermaids. Since then, they've been extremely mistrusting of a society that would stoop to accommodate monsters such as vampires and leave centaurs in the dirt.
5. According to centaurs, "all of time lays before [them], as a road lays before the traveler". They possess the ability to predict the future through the subtle movements of the stars. Unlike wizard Divination, a centaur's sense of the future is much vaguer, and tends to involve events with world-wide impact. For example, they were able to predict the rise of the Third Reich (and Grindewald) in the 1940s, constantly referring to "a power rising from the East" that would "devour the world, were it uncontested"; more recently, they predicted the Second Wizarding War, and were right about the peace between the wars being a mere interim.