This exercise is extremely beneficial: by keeping your hands closer together and your elbows against your sides, not only do you target your tricep muscles on the back of your arm, but you also engage the anterior (front) of your shoulders and your chest. By maintaining a perfect plank position with your back flat and glutes engaged, you also work your core. This exercise is extremely functional for multiple sports, such as surfing.
By all means, you do not have to do a tricep pushup on a medicine ball, as pictured above. This is a more advanced way of performing the exercise. The instability that the medicine ball provides gives an extra shock to your core which works even harder trying to keep your body in alignment.
To get into position for a tricep pushup without the medicine ball, get into a plank position, making sure that your wrists are directly below your shoulders. As you lower your chest towards the floor, keep your elbows next to your side, as opposed to bowing out as in a standard pushup. When you push your body back up, exhale and squeeze your abs, keep your elbows close to your side, and return to the starting position.
If you want to do a tricep pushup with the medicine ball, the directions are the same, except your hand position will be on the sides of the top of the ball. Still keep your elbows close to your side as you perform the movement. As always, keep your core engaged. As you get tired, your abs will save you from losing perfect form!!