There are many ways to do box jumps. The important thing when doing these is to land solidly on the box--your weight should be distributed over your whole foot, with both feet about shoulder width apart. You can technically just casually step down off the box, reset, then do the next solid box jump.

The key behind this is to land solidly. You want your weight to be evenly distributed over your whole foot when you land, rather than, for example, landing on the balls of your feet.

Throw the medicine ball as far as you can.  As soon as it lands, sprint to it and immediately do 10 pushups.  Repeat. 

Slam the medicine ball as hard as you can, making sure to use your whole body.  You should be squatting all the way down when you lift it back up, doing a front squat with it on your way up. 

Start by pulling up into the top of the pullup position. As you lower back down, simultaneously swing your legs and trunk as one unit. Repeat.

These are most commonly done on a basketball backboard. The video only demonstrates it on a pullup bar; you'll need to find something a little higher.
All you need to do is find something to reach for and jump up quickly in succession to touch your target object.  You won't squat for these--just jump from the ankles.

For this exercise you will do regular burpees, but instead of jumping for height at the end, you will jump for distance.  The pushups in the middle is optional, but it adds a distinct flavor that's hard to beat.

The objective of this exercise is to jump as high as you can at the top, and also to bring your knees up as high as possible in the "chambering" phase.  The default burpee for Fatal Fitness incorporates a pushup into the exercise.