My post count is low, but I'm no newbie so I'll help ya out a bit.

All Mountain entails all mountain trails. Lots of up and down steep hills. Very studly tires, don't want to slip on a root 95% of the way to the top. Usually middle-of-the-road for weight, can be FS or HT.

XC (cross-country) entails the same as above, but not on a mountain, and obstacles are probably not in the same tier of difficulty. Usually less-knobby tires, as the distances are longer and rolling resistance and weight becomes a larger factor. Freeride overlaps this largely, but generally shorter distances, more abusive trails (jumps to transitions, lots of splits between the "easy path" and "can I make it over that?"), and grippy tires.

DH bikes are the ones you take the lift to the top of the mountain with. Then pretend you forgot about the brakes and see how fast you can get down it. Weight is a non-issue as the true DH'ers pack on another 20lbs in armor, FS is mandatory to be considered DH, and there is usually a lot of travel in DH forks, the best of which are inverted for maximum travel (well over your standard ~80-120mm). These tires are huge (at least 2.4+), ideally tubeless with a low psi and some stans to fight off the pinch flats, since these guys literally roll over huge potholes at 30mph like it isn't even there going down the fireroads.

Geometry for every bike is different. You just need to feel what is right for you, and try to ride everything you can if you're considering a purchase. I mostly freeride, and I love my HT (especially for the climbs, it's a rocket uphill), but there are some downhills coming back to the trailhead that I wish I had a lower toptube on, as I would be quicker if I could get further over my back tire. A shorter stem (in the mail!) will alleviate some of this, for me. Specs and equipment is largely the same when it comes to the actual drivetrain across MTB's... the only real category with specific equipment is DH, which is always very heavy-duty stuff. Some DH'ers scrap quite a few gears as they aren't pushing themselves up anything and it's less stuff to break, which is an inevitability of DH'ing.

For my services I merely request your Teledyne. Please put some ultegras and pedals on, I'll look for it in the mail. Muahaha.