This is all governed by the equilibrium of the gravitational force produced by the earth and the centrifugal force produced by the velocity of any body flying around the earth. In that sense, the moon does not fall towards the the earth, but around it. A stable orbit is defined by the distance from the earth, where the gravitational force and the centrifugal force cancel out. The distance is dependent on the velocity of the body orbiting. The velocity of the Space Shuttle is chosen to be that large, in order to accomplish a stable orbit at the wanted altitude above the earth.

http://www.pic-upload.de/04.05.09/hqykqk.png
The above image shows you the result of the

3rd Keplerian Law for the solar system. If you translate the period P into an orbit velocity you get the following graph. It demonstrates the perfect alignment of the planets to the equation

,

where

is the orbital speed,

is the

universal gravitational constant,

is the mass of the central body (here: sun), and

is the distance of the orbiting object (here: planet) around the central body.

http://www.pic-upload.de/04.05.09/bgrc4o.png
If you do the same math for the moon and the shuttle around the earth, you will get the following results. Note that the Keplerian Laws are relative to the centre of the central body. So, to the altitude, you need to add the radius of the earth = 6378 km.

for the moon

for the shuttle

What prevents the Moon from falling to the ground like when I drop something on the floor?

The direction, the distance and the velocity. You can now calculate, what velocity something needs to orbit the earth at an altitude of 1 metre. See for yourself, and decide, whether this possible just by throwing.