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 (S, M, L, XL, etc) that allow for more growth, and can be used ambidextrously.
They are generally made of stiffer, duck cotton fabric which may be considered less comfortable and less flexible by some. They are also harder to put on and take off without help.
The real advantage to the back zip jacket would be to fencers who are considering fencing with both hands, while you are not allowed to change weapon hands during a single bout, you are permitted to change hands between different bouts in a tournament. A front zip jacket will be specific to the hand the fencer is using and would need to be changed along with the plastron and glove.
The front zip jackets are more comfortable and are easy to get in and out of. Many find them preferable when competing, 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.
An 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).
The front zip is measured by the chest size in inches or centimeters. They are much more specific to the size ordered and will tend to fit better, but are generally less forgiving to growth spurts and weight gain or loss.