Another style is the all purpose which has a deeper seat than a close contact saddle and was designed for both flatwork and jumping. It is kind of like a mix between the close contact and dressage saddle. Here is what they look like... http://www.fourwinds.net/esadcat/img/crossoft.jpg
You will also need a bridle, which is pretty self explanatory. Except your hand position is much different in english than western as you hold the reins with two hands in english.
Girths are the same things as cinches and buckle onto the billets on each side of the saddle. There are diff. Types of girths for dressage and close contact saddle and can be made of different material like fleece, neoprene, leather, etc.
Another thing you might find in english tack is martingales and breastplates. Martingales are used to control the horse's head carriage. The standing and running martingales prevent the horse from raising his head so high that he hits his rider in the face. A breastplate helps keep the saddle from slipping back.
I think that's basically all you need to know. If you need a more thorough explanation let me know.
Snaffle bit is any bit with no shanks for leverage. It can have a solid mouth, a joined mouth or a multi joined mouth.
A shanked bit is a leverage bit that utilizes a chin strap as well as the mouth piece for control. (The chin strap on this bit is attached to the rings on top - the same place as the headstall.)
A Pelham is an English bit that is meant to be used with two sets of reins. One set is attached the same as a snaffle and the second one is on the shanks like a leverage bit. The rider utilizes either set depending on what effect they want.
Those are just one example of each type. There are numerous variations of each.