Brakes

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.