Plyometric boxes are a fantastic workout tool and are common sight in Crossfit Gyms. They're quite expensive to purchase but much cheaper to make, and they aren't overly difficult to construct.
At my gym we've got a number of different boxes, several each of 12", 18", 20" and 24" heights. One of the other members told me that she wanted to progress past the 12" box she was using, but wasn't quite ready to move to the 18" box, and asked if I could make her a 15" to bridge the gap. For that reason these instructions will be based on a 15" box, but once you understand the basic construction you can easily make plans for any size that you'd like.
I'm also going to go into depth regarding setting up a workflow that lends itself to churning out multiple units. The woman that asked for this only wanted one, but as soon as I started working through the layout, I realized I could get four boxes out of a single 4' x 8' sheet of plywood. (Almost. I had some plywood laying around that I used to make two of the tops. Everything else came out of one sheet.)
The idea, though, is that if you're going to go through the trouble of making one, you might as well make a bunch, because it's only a little more effort. A lot of the work involves setting up jigs and templates, but once they're done you can just keep cranking them out.
I should also note that I have a pretty wide array of power tools. In most cases these just make things easier or faster, but many of them aren't a necessity. As I work through the instructions I'll try and point out how to adjust things if you don't have everything I do.
Step 1: Tools & Materials
Circular Saw & Guide - The first step will be to cut a 4' x 8' panel down into all the pieces you'll need to assemble your boxes. The more accurate your cuts, the better your box is going to fit together. At the bare minimum you'll need a good 8' straightedge to guide your saw. I use a guide system which consists of a pair of 50" aluminum extrusions that mate with a plastic base I've got attached to the base of my saw. A channel in the base rides in a raised portion of the guide so it tracks a straight line.
Drill/Countersink - I drilled and countersunk my holes with my cordless drill and then used a smaller pocket driver for my screws. A countersink is a pretty cheap investment to ward off splitting the ends of your plywood. Spend a few bucks to pick up one of these if you don't have one already.
Layout tools - At the very least you'll need a measuring tape. A decent square is handy as well.
Clamps - I'd almost call these necessary. I've got half a dozen quick clamps and I think I've used them in every single project I've ever worked on. They're almost like having a second set of hands. I could probably find a way to get along without them but it would be a lot harder.
Jigsaw - It helps to have handholds in your boxes to make them easier to move them around. You might get by drilling a couple of holes with a hole saw, but oval handholds are a little nicer.
Router - In my setup I made a template for my handholds, used the jigsaw to rough cut the holes and then used a pattern bit on the router to match them to the template. A faster way to work, but you could just use the jigsaw to cut the holes out.
Biscuit Joiner - A glued butt joint with screws is a pretty typical way to build these boxes, but biscuits add strength as well as keeping the sides perfectly aligned while you're assembling the boxes. If you have one, I'd use it.
Pocket Hole Jig - I used this for the tops so the screws would be hidden. In retrospect, I should have used these to join the sides as well, because then you wouldn't see any of the fasteners. Pocket screws actually make a stronger joint, because you're screwing into the face of the plywood instead of the end grain. If I make any more of these I'll go that route.
Orbital/Belt Sander - Now we're just getting into cosmetics. The boxes would be fine without these, but I'm anal retentive so I couldn't skip the finishing step.
Plywood - This is going to be your main cash outlay for the project, and the quality of the resulting box will depend on it. Hardwood plywood will get the best results. Particle board, MDF, OSB or any other sort of sheet good maybe be tempting but would be bad choices for this type of application. My local big box store had 4' x 8' sheets of maple ply for only $36, which is a steal. Shop around and you'll probably find a good deal.
Screws - Lots of people like to use drywall screws, but I prefer to use decking screws. I like Torx drive screws, and I've found the decking screws to be a little hardier than standard drywall screws. (Occasionally I'll snap the neck off one of those.) Personal preference, though. Use whatever you like, or already have on hand. I went with a 2" length.
Pocket Screws - If you've got a pocket hole jig you've probably already got these sitting around. Mine are 1 1/4" coarse thread Kreg screws designed for pocket holes.
Glue - Use a glue designed for use with wood. I like Titebond II.
Biscuits - If you're using a biscuit joiner, you'll need biscuits. I used size 20.
Polyurethane - I used this for a finish on the sides of the boxes. You could leave them unfinished, or put on whatever you'd like. The other ones at my gym were painted, but I opted to just stick with a clear coat. I had a few cans of spar urethane sitting around, so I went with that.
Top Material - You'll probably want something on the top other than smooth, bare plywood. There was some leftover textured paint from the construction of the others at my gym that I used. It's paint with an epoxy additive for wear resistance and a silica sand suspension for texture. When it's dried it's basically like sandpaper. The ones at the gym have seen a couple of years of constant use and they're still going strong, so this is a good choice. You could also get some rubber floor mats and cut those to size for the top and cement them on. If you go that route, make sure you calculate that into your height equation when you're laying out the boxes.