Brakes really fall into two camps- rim brakes or disc brakes. Within these
two you can take your pick from hydraulic or mechanical activation.
Rim brakes work by applying a pad to the rim part of the wheel. The friction
slows you down. Disc systems work on the same principal but the pad is applied
to a disc around the centre of the wheel. The advantage here is that mud and
buckles, which can play havoc with a rim brake, aren't much of an issue.
That said, a well set-up V-brake offers fantastic performance and may be all
you ever need.
The benefit of hydraulics over cables is that it won't stretch or need adjusting
and mud, grit and water won't get in and foul up performance.
Brakes are very important- if you go fast you've got to be able to stop, but
don't be tempted to shell out on discs or hydraulics unless you really need
them as they add weight, which can be justified by their performance- but
only if you need it.