Gravity is an attractive force. In other words, it makes things move towards each other. Everything which has a mass has a gravitational pull - even you and me - but the bigger (more massive) the object the larger it's gravitational pull. That is why if we drop a ball on the Moon it falls more slowly than it does on the Earth - the Moon is less massive and has a smaller gravitational pull.
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 Gavitational constant, r is the distance between the two objects and m1 and m2 are the masses of the objects.
Try the following animation to see what we mean.
You can also see how the effect of gravity changes on different planets in the Solar System, using the following tools: