There are many different types of bicycles out there. Here are some general categories with some basic information about each..
The Mountain Bike or All Terrain Bike is one of the most common types of bicycles. This is due to it's wide range of uses. Although they are built for trail riding and best suited for trail riding, a mountain bike (or ATB) can essentially be ridden anywhere. They will often have some sort of suspension to make rough riding easier on your body and the bike.
The Road Bike is designed for...you guessed it...road riding. They are built for speed and efficiency while riding on pavement. Think of a road bike as a sports car and a mountain bike as a truck. With a road bike you are fairly limited in where you can ride. Once you get off pavement you're in for some difficult riding. But on the road, there is nothing better. These types of bicycles will offer no suspension.
The Hybrid Bicycle is a cross between a mountain bike and a road bike. These types of bicycles typically won't have any suspension. With one of these bikes you will be getting a decent bike for off-road riding and a decent bike for road riding. This type of bike is well suited for the novice trail rider who also does a lot of road riding.
The Cyclo-cross Bicycle could probably fit into the hybrid category, although it is a different type of cross. This bike will look much like a road bike, but will be built with a stronger frame and stronger components. The bike is used for a mix of trail and road riding. It can handle some trail riding, but nothing very serious. They are often used for Hike and Bike races.
Cruisers are another type of bike. You may have seen one of these cruising around the beach. Mechanically, these bikes are very simple. These types of bicycles typically have one gear, no suspension, and foot breaks. They are not meant for trail or speed. They are meant for comfortable, leisurely riding.
Tandem Bicycles are a type of bike that I'm sure you've seen before. They tend to stand out a bit. These are the bikes that have two wheels and two seats and two riders. Basically, two people are riding the same bike. This is a very different type of ride for both riders. It can be a very rewarding experience in the correct situation. There are different types of tandem bikes that actually would put them into another category as well. There are mountain, road, and hybrid tandem bikes.
Recumbent Bicycles are ridden in a seated position with your feet out in front of you. You sit in a seat that looks like a chair. Depending on the type you steer with handle bars mounted about shoulder high or with bars below the seat. These can be powerer by foot or hand peddaling or both. They are very comfortable as they eliminate a lot of seat, neck, and back pain that can be a problem for some riders on conventional bicycles.
BMX Bicycles are smaller bikes, typically with 20" wheels compared to average mountain bike wheels at 26" and traditional road bikes at 700cc (27"). They have no gears and no suspension. Their uses include racing, jumping, and freestyle.