Off the cuff....

Cross Country - Think of Cross Country ski racing. Lightweight, nimble bikes that must climb well, handle technical terrain and intermediate hills. Front suspension forks (80mm - 100mm of travel) are the norm and new, lightweight F/S bikes have recently made their way into the sport as have lightweight mech. discs. This is what most folks think of when they hear "mountain biking" except for the folks who follow or participate in gravity racing events like DH and Slalom or Giant Slalom.

FreeRide - Basically what was a XC bike with a little more fork travel (>100mm), rear suspension, and disc brakes that will handle more "extreme" riding, i.e., bigger jumps, longer - faster downhills.

Downhill - Think of downhill (aka. Alpine) ski racing and the Downhill event. This is the gravity game -- where a heavy bike is not a disadvantage since you don't ride it up the hill, only down the hill as fast as your courage (or lack of common sense) will allow. LONGGGG travel front forks, LONGGG travel rear suspension, bomb-proof wheels, big disc brakes and low, laid back seating.

Slalom - Again, think apline skiing and the GS events. These are also gravity bikes, more like an aggressive Freeride machine than a pure downhill bike because they must be more nimble and maneuverable to handle banked curves, more pedalling sections on hills that aren't as steep and long as the DH courses.