Sorting Brake Calipers
There are two steps to sorting brakes. First, separate the two main types of brakes.
Caliper and Cantilever brakes attach to the bike differently. Caliper brakes attach above the wheel, and have a long bolt coming out of them. Cantilever and V-brakes attach next to the wheel, and are in two pieces. It's not hard to tell the difference, as you can see from these pictures: two types of caliper brake above (center-pull type to the left, side-pull to the right), and a V-brake brake below. A cantilever is similar to a V-brake in that it attaches to the same mounting studs, but it uses a straddle cable to pull the arms together instead of running the brake cable across the two arms.
Usually when you sort brakes they have the pads on, so caliper brakes will be one piece with two pads, and cantilever and V-brakes will be separate pieces with one pad each.
Sorting Caliper Brakes
There are two types of caliper brakes, shown in the picture below.