The jacket is an important piece of equipment that needs to fully cover the torso, groin, and arms. If you are buying a jacket there are two basic types: the front zip and the back zip. Again, you can find examples of both of these in the club's rental equipment. 

The back zip jacket is the most common beginner jacket. They are less expensive, come in simple to order sizes that allow for more growth, and can be used ambidextrously. They are generally made of stiffer, duck cotton fabric which is less comfortable and less flexible. They are also harder to put on and take off without help.

The front zip jackets are more comfortable and are easy to get in and out of, when competing they are preferable, because they make it easy to open and close to get some air in between bouts. The jackets are going to cost more and some of the tech fabrics tend to break down faster with wear and washing. Another important thing to consider, is that the front zip jackets are specific to the hand you use to fence. That is because the zipper must always be on the side of the body facing away from the opponent (so a right handed fencer will zip on the left side of the torso). While back zip jackets are sold in S, M, L, XL, etc... sizing, the front zip is measured by the chest size. They are much more specific to the size ordered, but are generally less forgiving to growth spurts and weight gain.