We see only one side of the moon because gravitational forces between the earth and the moon have locked the moon's rotational period to its orbital period. The moon turns exactly once on its axis each time it orbits the earth and we never get to see its backside.
The gravitational forces involved in this locking are known as tidal forces because they're also responsible for the earth's tides. They arise because when gravity acts on a huge object like the moon, that gravity isn't entirely uniform. Different parts of the moon are at different distances from the center of the earth and thus experience different strengths of the earth's gravity. These different strengths of gravity stress the moon and distort its shape slightly. That distortion lowers the total gravitational potential energy of the earth-moon system and creates an energy barrier that impedes the moon's rotation. It's a barrier in the sense that something would have to provide extra energy before the moon could turn a different side toward the earth, just as it takes extra energy to roll a bicycle wheel out of a pothole in the road. With no such extra energy available, this energy barrier locks the moon into a particularly orientation relative to the earth. Thus we keep seeing the same face of the slightly distorted moon, year after year.
The moon's gravity also distorts the earth very slightly, but the resulting energy barrier is too small to impede the earth's steady and rapid rotation. That's why the earth's rotation and orbit around the moon haven't locked together as they have for the moon. Nonetheless, the moon's gravity does create the earth's tides by distorting its oceans.
Answered by Lou A. Bloomfield of the University of Virginia.