There are two main categories of girths. Long girths and short girths. The category you have to choose depends on the length of the straps of your saddle. Most jumping saddles and all-purpose saddles have short girth straps, so you would need to use a long girth. Exception: All-purpose saddles with a focus on dressage. Dressage saddles have a long saddle flap and long girth straps, so you need to use a short girth (also called dressage girth).
This category is heading up to straight formed girths. Those girths are generally suitable for horses which are not very sensitive in the girth area and which also don't have any problems in this area (e. g. callus).
These girths have slight cutouts in the area of the elbow, so the movement of the horse's front legs are optimally ensured. It can also be wider in the middle thus causing the pressure from the girth to spread the whole length of the girth line.
This category includes girths with the stud guard, which is used in show jumping and eventing. These horses could get stomach wounds from the studs while jumping. Another feature of these girths is the extra fastener in the middle which is for attaching the auxiliary reins and breastplates. These straps can be taken off as well.
Nylon is a very sturdy and durable material which is also easy to maintain. For sensitive horses nylon might not be the favorite material because the girth might rub the horse's stomach when sweaty, which can cause sores.
Neoprene is a very soft and flexible material that is well tolerated and recommended for the use on sensitive horses. Also in combination with sweat it remains smooth. It's also easy to keep clean.
Leather is a natural product and requires more care. This means that it should be cleaned daily and very intensively every few weeks. You should use saddle soap and leather balm to keep the leather healthy. And it's an unbeatable material when it comes to comfort, durability and compatibility with (sensitive) horses.
Used especially for sensitive horses, whether on pressure or chafing, horse's can get relief by using a girth with lambskin. These paddings are available as girth tubes or they are directly attached to the girth. The advantage of girth tubes is that they can be washed separately. The advantage of a girth-padding-combination is that the padding can't slip around, so the risk of skin irritations are decreased immensely.