What is a Fixed-Gear Bicycle, and Why Would You Want One?
The oldest and simplest type of bicycle is the “fixed-gear” bicycle. This is a single-speed bike without a freewheel; that is, whenever the bike is in motion, the pedals will go around. You cannot coast on a fixed-gear machine.

Many enthusiastic cyclists ride such bicycles by choice, at least part of the time. Why would anybody do that? It is not easy to put into words. There is an almost mystical connection between a fixed-gear cyclist and bicycle, it feels like an extension of your body to a greater extent than does a freewheel-equipped machine. If you are an enthusiastic, vigorous cyclist, you really should give it a try.

There are many reasons, including: Fun, Fitness, Form, Feel & ‘Ficciency!

Fixed for Fun
It takes a bit of practice to become comfortable on a fixed gear. Most cyclists, trying it for the first time, will automatically try to coast once the bike gets up to a certain speed. The bike will not allow this, and it is disconcerting. It takes a couple of weeks of regular riding to unlearn the impulse to coast, and become at ease on a fixed gear.

It is worth going through this learning experience, however, because once you do so, you will discover a new joy in cycling. When you ride a fixed gear, you feel a closer communion with your bike and with the road. There is a purity and simplicity to the fixed-gear bicycle that can be quite seductive. Somehow, once you get past the unfamiliarity, it is just more fun than riding a bike with gears and a freewheel! If you won’t take my word for it, read some Fixed-Gear Testimonials.

Fixed for Feel
Fixed gear gives you a very direct feel for traction conditions on slippery surfaces. This makes them particularly suitable for riding in rainy or icy conditions.

This same feel for traction will help you learn exactly how hard you can apply your front brake without quite lifting the rear off the ground. Most fixed-gear riders only use a front brake–a rear brake is quite unnecessary on a fixed-gear machine.

Because you are more solidly connected to the bike, you have better control of it in bumpy conditions or in difficult corners.

Fixed for Fitness and Form
Riding a fixed gear on the road is excellent exercise. When you need to climb, you don’t need to think about when to change gears, because you don’t have that option. Instead, you know that you must just stand up and pedal, even though the gear is too high for maximum climbing efficiency. This makes you stronger.

If you have the option of gearing down and taking a hill at a slow pace, it is easy to yield to the temptation. When you ride a fixed gear, the need to push hard to get up the hills forces you to ride at a higher intensity than you otherwise might. Really steep hills may make you get off and walk, but the hills you are able to climb, you will climb substantially faster than you would on a geared bicycle.

When you descend, you can’t coast, but the gear is too low. This forces you to pedal at a faster cadence than you would choose on a multi-speed bicycle. High-cadence pedaling improves the suppleness of your legs. High rpm’s force you to learn to pedal in a smooth manner–if you don’t, you will bounce up and down in the saddle.

Most cyclists coast far too much. Riding a fixed-gear bike will break this pernicious habit. Coasting breaks up your rhythm and allows your legs to stiffen up. Keeping your legs in motion keeps the muscles supple, and promotes good circulation.

On any road bike, the rider must learn to un-weight the saddle to ride over bumps. Most cyclists coast to do this. A fixed-gear rider will learn to “post” over bumps without breaking stride.

Fixed for (e)Fficiency
A fixed-gear bike is considerably lighter than a multi-speed bike of comparable quality, due to the abscence of the rear brake, derailers, shift levers, and extra sprockets. A fixed-gear bike also has a substantially shorter chain.

A properly set-up fixed gear has a perfectly straight chainline. This, plus the abscence of derailer pulleys, makes a real improvement in the drive-train efficiency, an improvement you can feel.

The following article is taken from the late  Godfather of Modern fixed gear riding Sheldon Brown and from his page: http://www.sheldonbrown.com/fixed.html