A center square is.... hard to describe without pictures.
Let's start with what it does. Let's say you have a finished dowel, of wood or metal, and you want to mark the center point on the end -- how do you do it? Well, you use a center square, a.k.a. a center finding gauge or radial square.
Theoretically, you could do this by having a right angle that set over the curve of the piece, and having a straight edge that precisely bisected that angle, but was raised up to pass over the end of the piece. The straight edge will scribe a ray across the diameter, passing over the center of the circle, and drawing two of these that intersect will give you an X that marks the spot.
Pretty much anything you could design that does that, is a center square. The prototypical center square is a disk with a wedge cut out of it, with a straight edge welded to lie flat along the top of the disk, intersecting the wedge along the disk's diameter (don't bother to re-read that; just look at this). In practice, most center squares are actually just a right angle made of steel with a bisecting bar that clamps to lie above the L. One of the most common alternate heads for a combination square is just this L bracket, and that's what most non-specialists use.