Gravity is a force pulling together all matter (which is anything you can physically touch). The more matter, the more gravity, so things that have a lot of matter such as planets and moons and stars pull more strongly.
Mass is how we measure the amount of matter in something. The more massive something is, the more of a gravitational pull it exerts. As we walk on the surface of the Earth, it pulls on us, and we pull back. But since the Earth is so much more massive than we are, the pull from us is not strong enough to move the Earth, while the pull from the Earth can make us fall flat on our faces.
In addition to depending on the amount of mass, gravity also depends on how far you are from something. This is why we are stuck to the surface of the Earth instead of being pulled off into the Sun, which has many more times the gravity of the Earth.