So completely random thought but how would a centaur take the cavalier fighter subclass? Would it need a mount or would it just be able to fight using its horse like build? And if not, how would it even ride a horse?

You don't actually need to ride a mount in order to be a cavalier.  Heck the subclass only has one ability that's related to riding.  There's no reason a centaur couldn't become a cavalier.  As for riding horses, there's technically no restriction against it but it would look darned silly, plus the weight of the centaur plus equipment plus any barding the horse might have would be a bit much for all but the largest and strongest horses.

Nah, what ya really wanna do is be a Centaur Rune Knight with a Cavalier in the party to ride you when you grow to Large size. Either that or have a Kobold Cavalier in the party and then the Centaur can be a Paladin. 😉

This is something worth asking your DM about, and seeing if they would homerule to allow a Centaur to receive the cavalier mounted bonuses without a mount. It might not be be proper rules as written, but it is one of those things a DM very likely would allow—the alternative creates some rather ridiculous situations. Personally, I am letting one of my players do this and it really is not causing any trouble.

As others have pointed out, all but one of the Cavalier's features work mounted or dismounted.  My instinct tells me to allow Born to the Saddle to have advantage on saves where the centaur risks otherwise be knocked prone, but it's not a tested hunch.  

I, like many DMs I'm guessing, don't allow centaurs to have traditional mounts.  Aerial and aquatic mounts I haven't had the opportunity to rule on, I'd probably require specially constructed saddles for such an arrangement.