What happens if you crumple up the piece of paper into a ball?  When you decrease the surface area of the paper by making it into a ball then friction from air does not affect it as much.  So when you drop a ball of paper, the only force acting on it is the force of gravity, therefore it will fall as fast a piece of rock.  This animation shows you that.  But don't take our word for it.  Try it for yourself.

Here is a movie clip that shows Astronaut David Scott doing a similar experiment on the Moon where there is no atmosphere so that the fall of a feather and a hammer will only be affected by gravity of the moon.

Here is an animation that shows you how the gravity of Earth keeps the moon in orbit around Earth.

Extra! Extra!

You may have noticed from reading "Why Can't I Jump Very High?" that any thing with mass has gravity. This includes you and your friend. This means that if you two were standing next to each other, you and your friend would be pulled towards each other with a certain amount of force. But because your masses are so small compared with something like Earth, the force of gravity between you two is too small to be noticed.
Galileo was the first person to show that different things fell at the same rate on Earth, but he could not explain why, or if this was true all over the universe.

Links

From Apples to Orbits: The Gravity Story - A website created by students about gravity, it's history, and games that will sure suck you in.
Gravity Song - A short cute song about gravity.
Gravity Question - How far do you have to go to escape Earth's gravity?
Hitting Homeruns On The Moon - Find out how certain sports will change if you were playing them on the moon, which has less gravity than Earth.
Microgravity: Always A Bad Hair Day - Shows and explains the presence of the small gravitational force of Earth on astronauts in space.
Your Weight On Other Planets - Find out how the gravity of other planets affect your weight on them.
Soda Play - Create your own virtual toy with masses and springs, and see how they behave under different values for gravity. (Advanced)