There are three main types of bikes based on suspension:
Rigid Frame
Hard Tail
Soft Tail
Rigid Frame
Fully rigid frames are becoming rarer as technology advances however they are still a popular choice among mountain biking enthusiasts. Most Road and cyclocross bikes can also be considered fully rigid frames as well. Without the added weight of suspension systems the frame leaves the rider with a considerably lighter bike than its suspension equipped counterparts. Not only this but rigid frames also offer greater feedback and ability to feel terrain beneath the rider as no suspension systems are present to dampen interaction/contact between the rider and the trail. Lastly rigid framed bikes are cheaper and easier to service than most other bikes because there are no suspension systems present. Despite this rigid framed bikes have their obvious downsides; mainly being comfort. Since no suspension is present it is entirely up to the rider to dampen impacts through body position and their joints thus increasing fatigue and potential for injury.
Hard Tail
Hard Tail frames are one of the most popular forms of mountain bike on the market. This is simply because they are relatively cheap and lightweight while still providing the benefit of having suspension. One of the greatest benefits of a hard tail bike is that during a climb they are very efficent at transferring energy to the pedals due to the fact that all of your force is absorbed by the pedals rather than the suspension. This enables the rider to ascent faster with less wasted effort. Hard tail bikes are generally cheaper because since they have been around for longer you get more for what you pay as research on full suspension is relatively new and more expensive than hard tail. For most beginner to intermediate riders the hard tail is a great place to start, especially if the buyer is limited by cost.
Soft Tail
For all riders a soft tail frame is the peak of comfort and smooth riding. The only catch is the cost. Due to the fact that soft-tail bikes are newer in concept than hardtails the consumer ends up paying more due to licensing and patents. Soft tails are superior to hard tail and rigid framed bikes due to their smooth ride and ability to carry speed over rougher and more technical sections of trail. They can also be beneficial to newer riders, children and elderly riders due to the lessened impact and easier riding style. They are also the frame of choice for downhill riders as a hard tail or rigid frame simply wouldn't be capable of carrying as much speed over bumps as well as the fact that the added weight would add to the riders momentum and speed.
Unique and Niche Types
Singlespeed
Singlespeed bikes have recently gained a cult following in the mountain bike community and vairous events have been orgnaised exclusisevly for single speed bikes. These bikes are just like any other bike apart from the fact they one have one gear. The reason for this is the increased simplicity and challenge of only have one gear to work with. The practical advantage of course is that you lose less power to the transmission and added weight of all those gears. Not only this, but a singlespeed bike will make you a better and fitter rider as you learn to conserve momentum and speed on climbs and technical sections These bikes have great character and are very unqiue. The only downside to riding these bikes is the potential for knee pains as a result of a higher gear ratio.
Unicyle
Unicycles are potentially one of the most unique and rarest sights on your local trail. These one wheeled contraptions are one of the wackiest choices and only the most skilled riders will be able to master these. Believe it or not the unicycle is ridden at major events like Karapoti Classic and it's riders are able to hold their own against competitors armed with regular bikes with two wheels. Mountain unicycles have thicker, wider tires for shock absorption. Many riders choose to use long cranks to increase power when riding up hills and over rough terrain. A disk brake is sometimes used for descents, the brake handle is attached to the underside of the handle on the front of the saddle.