Super is short for superstructure, which refers to the boxes placed on a beehive for bees to store honey. Historically, a super was always medium, 6 5/8-inch tall box or a shallow 5 3/4-inch tall box. Both are traditionally referred to as supers, exclusive of the deep 9 5/8-inch box used on the bottom of the hive.
Since the word super is short for superstructure, it could actually mean any size box that is extra and in addition to the "365 hive". This is what we mean by that, there are a certain amount of boxes a beekeeper leaves for the bees to live in for 365 days a year. The boxes remain year around on your hive is your "structure". Only during times of lots of nectar collection, are additional boxes added which would be your "super"structure. Whatever the size box used, deep, medium, shallow, 10-frame, 8-frame or 5-frame, these boxes can be considered supers.
In recent years, it is more common for beekeepers to not use deep boxes and use medium boxes exclusively for their whole hives (Read more here). The size of the box does not dictate what a bee will use it for. Bees can use any size box for any purpose they need. A beekeeper who has been keeping bees for a long time may refer to a medium box as a super, regardless if it is being used for brood or for honey. This is only out of habit because they may have started beekeeping before anyone had the idea of using medium boxes for the whole hive.
If you want to read more about if you should or should not use all medium boxes or a combination of box sizes, click here.