Gravity does one thing, it pulls objects together. In the case of the earth, the pull is massive, and the force of gravity on at the surface of the earth can accelerate any body downwards at a rate of 32 feet per second per second. You don't need to worry about the calculations, but understanding how to exploit this force can improve your martial skills.
There are a number of ways that gravity can be made to work for us, and against our opponent. Gravity can:
Add Power to Your Attacks
Although the force of gravity pulls you down, you can exploit this force to create force horizontally, and even create force upwards.
Downwards force is easy enough. You can pull down with your core muscles to pull against your mass, and bring your opponent down. You can also bend over at the waist to add to this force. You can also push down from above, by using the force of gravity. We call this "slumping."
Horizontal force is a little harder to understand. How can a downwards force impart horizontal motion? In order to understand this, you have to understand friction. Gravity causes our weight to push our feet into the ground. There, the stickiness of the feet allows us to push horizontally, and also turn to create angular momentum. Without gravity, we would not have a base to push against.
There is another mechanism related to this. By dropping our weight, while our feet are anchored due to friction, and our structure maintains integrity, we can push horizontally against an object. As long as our feet don't slide, and our structure does not break, the drop of our body mass can be translated into a horizontal force where we contact our opponent. This is a little hard to explain in words, but easy enough to demonstrate.
Creating force upwards is even more counter-intuitive. It depends on the ground reaction force (Newton's Law) and the elasticity of the tissues of the body. By dropping quickly, you can store energy in the leg tendons (for instance) as elastic potential energy. As long as your timing is right, you can return this energy for attacking. In effect, you bounce off the ground.
By bending the legs fast enough, you duck with the acceleration due to gravity. This is the fastest that anyone can duck.
Off-balance Your Opponent
The force that you generate can be used to hit, but also can be used to misalign your opponent, to off-balance and break his structural integrity. This makes his defence more difficult, and may permit you to take him right to the ground.
Smash the Opponent to the Ground
This should be fairy self-explanatory, but the ground can be a better weapon than your body. Also, the force with which a falling opponent hits the ground can be considerable. You can add to this force by using your weight to take the opponent down harder.