Single Speed: Fixed Gear or Freewheel? Choosing a Riding Style
The bike frame you have influences what wheel you need for it. However there is also a seperate question about what kind of bike you want to be riding which also must be answered in order for you to know which wheel you need.
The activity of single speed cycling involves two different types of cycling, corresponding to two different kinds of single speed bicycles. You need to decide which type of bike you want. (Though some set ups can allow both kinds of riding).
A freewheel single speed bicycle operates in the same way as a standard multi-speed road or mountain bike, the only difference is that you are always stuck in the same gear! Everything else is as normal, most specifically, you can still freewheel – stop pedalling while the bike is in motion and it will continue to move.
A fixed gear single speed bicycle operates in a different way from a standard bike. It is also stuck in one gear ratio just like the freewheel single speed. However, this bicycle will not allow you to freewheel. The absence of a freewheel mechanism in the rear end of the bike means that any time the rear wheel is moving – forward or back – the pedals will be rotating in a fixed ratio with that motion. And if the pedals are moving - forward or back - the same will be the case for the wheel.
So, on the contrary to a freewheel bicycle, if you try to stop pedaling when you're in motion you will find that the pedals keep moving but your reverse pressure slows the bike down. If you ever managed to pedal backwards you would be cycling backwards!
You will find discussion of the different perceived benefits of these two ways of travelling amongst many cyclists. A quick summary of the reasons you might consider fixed gear: it forces you to pedal throughout your journey, good for training yourself up and getting fit; some feel that it makes cycling feel more immediate, like you are yourself a part of the bicycle, even a part of the road(?); some people prefer using their feet on the pedals of a fixed gear bike as a braking mechanism to the standard rear hand brake on other bikes - a question of personal preference...