Gravity is a natural force of attraction between two physical objects. The gravitational force between two objects is directly proportional to the masses of the objects and inversely proportional to the distance between them. In other words, the more massive the objects, the stronger the gravity. Also, the further the two objects move apart, the smaller the gravitational effect becomes. So even something as simple as two pencils lying next to one another on a table each exert a gravitational force on each other. Because the pencils are so small in mass, however, this gravitational force is incredibly small and essentially negligible. As far as why gravity exists, it is quite complicated since it involves the theory of relativity and the curvature of spacetime.
In terms of the Earth, gravity is what keeps all objects on the surface of the planet since there is a gravitational force between the planet and every object existing on it. Just as the Earth exerts a gravitational force on a person, the person also exerts a gravitational force on the Earth. But because the Earth has so much more mass than a single person, its gravitational force is so much stronger and the only one actually observed.