Gravity is a fundamental property of matter, all objects with mass have gravity, and all objects with mass are effected by gravity. Further more the more mass an object has, the more gravitational force it exerts on other objects. The Earth, being a massive object, exerts enough gravitational force to keep us all stuck to its surface (except for a very lucky few astronauts), and keeps the moon in orbit. The Sun's gravitational force keeps us in orbit around the sun.
You are most likely sitting down. Do you feel the chair pressing against you? The chair is resisting force of gravity that the Earth is applying to you, and keeping you in one place. Should the chair be removed, you will fall to the ground because of the force of gravity pulling down on you.
Astronauts are also experience about as much force from Earth's gravity as we are on the surface. How do they not fall down to Earth? The answer is that they are constantly falling towards the earth- but are moving so fast that they keep missing! This speed required to consistently miss the earth is called the 'escape velocity'.
Try finding two similarly sized, but different weighted object. Hold them out one in each hand. Before you drop them, have a guess which do you think will land fist?Now drop them at exactly the same time. You might be surprised to notice that both land at the same time (any difference will be from wind resistance). The reason is a bit technical, but the laws of physics are such that the mass divides out perfectly causing all objects, regardless of mass, to accelerate at the same rate under gravitational forces. All objects on earth will accelerate down at 9.8meters per second squared.