Why does my fencing jacket zip up the side instead of the middle, and what is that kind of jacket called?

It is simply called a front-zip jacket (plenty of fencers say side-zip), and the zip runs down the side away from your weapon arm on purpose. In the en garde stance your weapon-arm side faces your opponent and your other side is turned away, so the opening sits where their point rarely arrives. The rules back this up: clothing must have no openings where a point could catch, and international rules add that the zip must sit under a flap that overlaps toward your weapon arm, so a blade sliding across your chest rides over the seam instead of into it. That is also why every front-zip jacket is either right-handed or left-handed.

The other style you will meet is the back-zip jacket, which closes down the spine. Nothing opens on either side, so one jacket fits right-handers and left-handers alike, and clubs often buy them for loaner gear instead of stocking two versions of every size. The catch is that you need a friend to zip you in and out. Most fencers move to their own front-zip jacket as soon as they buy whites, and it is what nearly everyone wears at tournaments.

The plastron, or underarm protector, is the half-jacket worn underneath. It covers the weapon-arm side of your chest and armpit with a sleeve down to the elbow, because that side faces the opponent and the armpit is where a blade can slip past the jacket. USA Fencing requires one with every uniform, with no seam in the armpit. A hard chest protector is a separate extra piece, required for women, and it never replaces the plastron.

When you buy, order everything for the hand that holds your weapon. A left-hander needs a left-handed jacket and plastron, plus a left-handed lamé for foil or sabre, and some knickers are handed too, so check the listing. If you head to national events later, your last name must go on the back of the jacket or lamé (or on the rear thigh) in capitals 8 to 10 cm tall, which is easy to add to a front-zip jacket.