You are correct -- the pull of the moon causes tides. All objects have at least a tiny bit of gravitational pull, but bigger objects (with a larger mass) have bigger pull. The other factor is distance -- the further away something is, the smaller force it exerts.

To compare, the sun also causes a slight tide on earth. The mass of the sun is so big that it keeps earth in orbit even though it's so far away. On top of that, we can feel it's pull on the earth through the tides. When the moon and sun align on one side of the planet, both of them are pulling in the same direction and we get stronger tides (called spring tides). When the moon and the sun are on opposite sides of the earth, we get smaller tides (called neap tides).

The center of the earth is much closer to our oceans than either of the sun or the moon, which is why we feel the earth's gravity the strongest, but the tides are a good way of seeing how gravity from other bodies affects us.

I took this picture from www.atlantickayaktours.com

Anything exerts tides, no matter what other gravitational objects there are. The fact that the Earth has gravity is what prevents the water from going all of the way to the moon, though, and keeps it on the surface of the Earth. Think about it this way: the tides are a few feet each day, but how deep is the ocean? That gives you an idea of the relative importance of the Earth's gravity to the moon's in determining where the oceans go.

Great question! You are correct in saying that Earth's gravitational pull is greater than that of the moon. However, the water at the surface of the Earth right underneath the moon is actually closer to the moon that it is to the center of the Earth (it's about 4000 miles closer). Therefore, surface oceans 'feel' the moon's gravitational pull slightly more than that of the Earth's pull. However, I want to point out that this difference is not much, and that's why we don't see our oceans flying into space.