Although the pull of the Earth is stronger than that of the Moon - we still feel the effect of the gravitational pull of the Moon here on Earth, especially through the rise and fall of the oceans.
It is gravity which holds us down on the Earth, and keeps the Earth in orbit around the Sun. As well as increasing mass leading to a stronger gravitational pull, it also gets stronger the closer the object (or objects) are, this is why planets closer to the Sun move faster than those further away. It's also why the Moon orbits around the Earth first (which is close) then the Sun which is much more massive.
We can work out the strength of the gravitational force using the equation below:
F=G (m1m2) / r2
Where F is the Force, G is called the Gravitational constant, r is the distance between the two objects and m1 and m2 are the masses of the objects.
You can see how the effect of gravity changes on different planets in the Solar System, using the following tools:
Gravity Simulator - use your mouse to play with the ball (see image to the left).
Gravity Workshop - A classroom based lesson where the students use the simulator.