There are two main types of riding helmets: fixed peak hats and skull caps. Skull caps are domed helmets, which are usually worn with a hat silk. fixed peak hats are usually coated in blue or black velvet and have the peak as part of of riding helmet. As a rule, riders will wear skull caps for day to day use. The only time that people really wear the fixed peak varieties is when competing in horse shows.
The reason why it is important to wear riding helmets is simple: safety. If you do fall from your horse, you really do want all the protection you can get.