Gravity is something that exists all over the universe. It causes an attractive force that draws everything towards everything else. If you hold a pen up in the air and let it go it will fall downwards, towards the Earth. Gravity is directly proportional to the mass of the object; if the mass of object A is twice that of object B then the gravity of B is twice that of the gravity of A. This is why the pen falls towards the Earth and not the other way round, it's because the Earth is massive compared to the pen and exerts more gravitational pull. In actual fact the Earth does move slightly towards the pen but very little as it's mass is too massive for the gravity of the pen to move it much.
The strength of gravity is also inversely proportional to the square of the distance. This means it gets weaker as you get further away from the object. Using this rule gravity would be four times weaker at twice the distance, because two multiplied by two equals four, and sixteen times weaker at four times the distance (the square of four is sixteen).
The Force of Gravity is measured in Newtons and it creates an acceleration of 9.81 m/s2. Every object falls at the same rate. This isn't apparent on Earth because air resistance alters the speed at which an object falls. On Earth a feather would fall much more slowly than a hammer. On the Moon, because there is no air, a feather falls as fast as the hammer and will hit the ground at the same time. Galileo predicted that two objects, regardless of their different masses, would fall at the same speed. This was confirmed in a famous experiment conducted by astronauts David Scott and Jim Irwin as part of the Apollo 15 mission during 1971. Below is NASA video footage of the event.