Yes, hunters is a style of riding, if you will. There are three basic types that you will likely see at a show.
Sometimes hunter equitation is offered. Hunter judges look for a horse that is suitable for a hunt, hypothetically. Really, they're looking for a collected, rhythmic round with good distances, lead changes, nice form of the horse over the fence, willingness and obedience in the horse, basically, a pretty round. It is not timed and you are judged subjectively. Rails down will count against you but aren't assigned a penalty value. Hunter courses are generally very simple. Most little girls in the US start in hunters. Turnout is generally conservative but is usually fairly relaxed at local or schooling shows. You'll see more colorful show shirts and sometimes hunter green or grey jackets instead of just navy and black. A standing martingale is allowed for over fences classes but prohibited on the flat.
Equitation is judged on the riders position and handling of the horse. These courses are not timed but are generally more complicated and can include roll back turns, trot fences, hand gallop fences, narrow fences, combinations, etc. A rail down does not necessarily mean you're out of the ribbons unless the judge decides the rail was the fault of the rider and not the horse. The biggest equitation class in the US that most people have heard of is the MaClay finals at the national horse show. Turnout is important and very conservative. Braided horses, navy coat, white shirt, etc. No flashy colors, no bedazzled brow bands, nothing that makes you pop out. Your excellent riding should attract the judges attention.
Jumpers classes are based purely on speed and accuracy. It doesn't matter what you look like (although you should still dress nicely out of respect for the sport and the judge) or your position as long as you go fast and clean. Clean means no rails down and no refusals. Each jumping fault adds four penalties. Many jumper riders wear a nice polo tucked in with breeches and tall boots. Some wear a jacket too but it's not necessary until you are at the upper levels. You can also leave your hair out of your helmet. This is the type of jumping you see at the Olympics.
Hunters and Equitation are usually offered as a division meaning there are some over fences classes and some under saddle classes. There are different requirements for each, such as the standing martingale for hunters. Check your prize list for details. In higher level shows different types of hunters will be offered such as performance, handy, and show hunters.