XC (Cross Country) - Riding along hilly trails where some bumps and smaller obstacles, such as rocks, roots, or depressions, may be encountered. XC bikes tend to go very fast on smooth terrain and can climb very well, but will have a litte trouble on descents.
AM/Trail (All Mountain) - Similar to XC but also involves moderate to steep slopes, moderate obstacles, and small jumps. AM bikes are stronger and heavier than XC bikes, but can handle more agressive trails while still able to climb and ride the smooth stuff fairly well.
FR (Freeride) - This riding style is for skilled riders and involves aggressive slopes, large obstacles, and jumps. These bikes are designed for the most agressive terrain. They can't climb very well, but can do some climbing when necessary.
DH (Downhill) - This riding style is for highly skilled riders and involves many of the things you see in Freeride except at really high speeds. Forget about doing any climbing with these bikes.
DJ (Dirt Jumping/Urban) - This style of riding is for highly skilled riders and involves large jumps and drops off dirt jumps or urban terrain or used at skate parks.
4X (Four Cross) - This style of riding is similar to BMX racing except with mountain bikes. These bikes are designed specifically to be raced on a 4X
These are your basic types of mountain bikes and types of riding. You'll see different terms and many bikes can blur the lines between styles. You will definitely see lines blurred between XC-AM, AM-FR, and FR-DH. Basically you just figure out how aggressive you want to be and get the bike that fits.