Q: Why is there more gravity on earth than there is on the moon? Does it
have something to do with the size of the moon?

A: You have the correct idea. The force of gravity increases as the mass of the object increases. It also falls off rapidly with your distance. The force of gravity on the surface of the moon is about 1/6th of the force of gravity on the Earth. The moon is actually about 80 times less massive, but because it is also much smaller, when you're on its surface you are much closer to its center. So the combination of being much less massive and much smaller leads to the difference.

For more on the gravitational strength of planets see Small Scale Effects Page 4 or try this Java applet (requires a Java enabled browser).