XC Cross Country
Sometimes referred to as "trail", these bikes are typically fairly lightweight and feature neutral geometry that allows for a very versatile end use. They're good climbers due to their weight, and good at general downhill due to 4-5 inches of travel. A very good "do it all" bike. Available in hardtail or full suspension.
These bikes are designed for cross country mountain bike racing and feature super light weights, shorter wheelbases and steeper steering angles for quicker turning. 2-3 inches of travel have a stiffer ride but make the bike more efficient for pedaling during a race situation. The fit is usually low hands and long reach for efficient climbing, and an overall end use oriented more towards speed than versatility. Often times people say these bikes are more "twitchy" and rougher to ride than XC-non race bikes. Hardtails are still a favorite among racers, but the full suspension bikes are way more efficient pedalers than in the past and beat you up a lot less.
A bike with 29 inch wheels rather than the standard 26 wheels is said to roll easier, has a higher clearance for clearing obstacles and is quickly becoming a favorite among racers and fun riders. Picture you and I have wheelbarrows... mine has a real big wheel and yours has a small wheel. Which one rolls over bumps and obstacles easier? Get it now? Try one and see what it feels like!
This type of bike is a medium weight, neutral geometry, bigger travel (5-6 inches) that is better at downhills, big obstacles or jumps. A little tougher to pedal up the hills but more confident down them. As technology lightens up bigger travel bikes and makes them more efficient to pedal, this category is expected to grow.
A bike with no rear shock or rear suspension. Harsher to ride but more efficient to pedal. The bike bobs less and every pedal stroke puts power to the rear wheel rather than being absorbed slightly by the suspension. A high end hardtail will almost always be lighter and cost less than an equally equipped full suspension due to less rear componentry.
Front and rear suspension! A much more forgiving bike to be ride for any distance. Today's "pro-pedal" features are making full suspension bikes more and more efficient to pedal (i.e., less "bob"). Since the rear wheel moves instead of the whole bike when riding over bumps or obstacles, the bike dances around less and is said to be more controllable. Rear suspension will add some dollars to the price and some weight, but most people feel they are rewarded greatly by having it.