It wasn't always this way with the moon -- the moon _used_ to rotate relative to earth, so that an observer on earth would have seen different faces of the moon as it spun. But it gradually stopped spinning due to something called "tidal locking".

Here's how it works:

You know that the moon causes the tides in the ocean, right? The moon's gravity pulls on the earth, and it pulls most strongly on the face of the earth that is facing the moon. The land on earth doesn't particularly care about this extra tug, but the oceans do. Water is "lifted" towards the moon, and flows to make a bulge that faces the moon. (There is a bulge in the back of Earth too, pointing away, which is related.) As the earth turns, this "bulge" flows through the oceans, always approximately facing the moon; we see it as tides moving up and down.

Well, the earth does the same thing back to the moon -- tidal forces from the Earth are about 80 times stronger than the moon's tidal forces on us (because the Earth mass is 80x that of the moon). However, there are no oceans on the moon -- so no liquid sloshes around like it does on earth. Believe it or not, the earth's tidal forces _do_ deform the moon itself, though, ever so slightly. Back when the moon used to rotate relative to us, there was a little "land bulge" on the moon's surface, which "wants" to face the earth. When the moon used to rotate, though, the rotation would tend to carry the "bulge" along with it. This set up a tug of war -- the moon's rotation pulls the bulge away, and the earth pulls back on the bulge, against the rotation. This, then, basically acted like a bicycle brake: the earth's tidal forces constantly acted to pull against the rotation, hereby slowing the rotation down until it stopped. So now -- the "bulge" just points right at us. It's not very big -- but it is still there.

The moon is not the only heavenly body that is tidally locked. Mercury is tidally locked to the sun -- meaning if you lived on Mercury, it would either always be day or always be night, depending on which side of Mercury you lived on.

A longer description can be found at
tidal-locking

They give an estimate for the time it takes for the locking to happen -- my quick math said something like 300,000 years for the moon. Check my math -- I wasn't very careful about it, and may be completely wrong!

The moon revolves around the earth AND also ROTATES on its axis once per month.

So you can do an experiment or demonstration. Imagine the MOON did NOT rotate on its own axis (that is that the moon did not spin around its own axis but ONLY REVOLVED around the Earth. If you do this using a simple demonstration you will note that if THAT was the case, then we would see the entire moon after one revolution.

Now do the demonstration again, but this time allow the moon to rotate on its axis in exactly the same amount of time it takes the moon to revolve around the Earth...you can get a friend plus you as earth and actually demonstrate this to convince yourself.

What a great question! The Moon rotates about its axis every 29.5 days(just like the Earth rotates around its axis every day)--this is why the moon has phases, as different parts of its surface are exposed to the Sun over the course of a month. It also takes the Moon 29.5 days to complete its orbit around the Earth (this is relative to the Earth--from the Sun's perspective, it only takes 27.3 days for the Moon to complete its orbit, but during this time the Earth has been moving along in its orbit around the Sun, and the Moon needs an extracouple of days to "catch up"). Because the time it takes for the Moonto orbit the Earth is the same as it takes for it to rotate about its axis, we always see the same face of the Moon. (You can try this out for yourself--if you walk in a circle around a friend, you will have to rotate at the same rate as you are walking if you want to stay facing your friend.)

It's not just a coincidence that these two things happen at the same rate--we think that a long time ago the Moon actually rotated more quickly than it orbited, so if we had been around then, we would have been able to see different faces of the Moon. The Moon's rotations lowed down, however, due to what are called tidal forces: the side of the Moon that's closer to the Earth feels a slightly stronger pull due to the Earth's gravity than does the side of the Moon away from the Earth, so it deforms a little bit--it gets a little bit longer in the direction facing the Earth. As different sides of the Moon faced the Earth, different parts of the Moon would get deformed, and all of this deformation produced a lot of friction. This friction slowed down the Moon's rotation over a long time, until eventually the same side always faced the Earth, and the deformed part did not have to move anymore.

The Moon's gravity does the exact same thing to the Earth: the waters in the ocean on the side of the Earth that faces the Moon are more strongly attracted to the Moon than are those part-way round the Earth, which in turn are more strongly attracted to the Moon than those that are on the opposite side of the Earth. This is what produces the cycle of low and high tides that you can see in the ocean. The friction of these tides is slowing down the Earth's rotation as well--in the past the Earth rotated about is axis much more quickly, so billions of years ago, days were only a few hours long.

Tides - the moon pulls on the side of the Earth closest to it more strongly than it pulls on the opposite side of the Earth, which effectively stretches the Earth as it is being pulled on from one side but not the other and the Earth is ever-so-slightly flexible. This is what causes the ocean to come up on land on the side facing the moon and the side being (relatively) pulled away, because water is more fluid and flexible than land is, because it is liquid. However, this also exerts a pull on the Earth against the Earth's rotation, just as if you pull on a long projection on something, it will rotate toward the direction you are pulling (imagine pulling on the end of a spoon sitting on the table - the body of the spoon will rotate to be opposite the direction you're pulling). But the Earth is already spinning on its own axis, so the moon's tides are actually slowing the Earth's rotation. This is very slow, but 400 million years ago, the year had about 400 days instead of 365.

Just as the moon exerts tides on the Earth, so does the Earth exert tides on the moon, except the moon has no oceans to get sloshed up and down as this happens. Because the Earth is so much more massive than the moon, the moon inevitably loses more angular momentum in its own rotation. Put simply, it has become tidally locked, with its rotation period being the same as its orbital period, because the Earth is basically dragging it along as it orbits the planet. Because one side of the moon happens to be being pulled on by the Earth, just like the spoon on the table, that side follows the Earth around as it is pulled, and always stays pointed toward the Earth. Thus, we never see the far side of the moon.

A way to visualize why we only see one side of the moon is to walk in a circle while always facing the middle of the circle. While it may not feel like it, you are actually rotating your body while revolving around the circle. The key here is that the period of revolution is equal to the period of rotation. When you have walked halfway around the circle, you face the exact opposite direction compared to when you started. You complete 1/2 revolution in 1/2 rotation, meaning the same rate of revolution and rotation.

The moon's revolution and rotation periods are about 29.5 days. If you want to model the earth and moon together, have a friend stand in the middle of the circle and spin in place 29.5 times for every time you complete a full circle as the moon. Just don't get too dizzy! Your friend in the middle will see you "rising" and "setting" each time he spins in place. This is why we see the moon rise and set each day--the earth spins in place once a day, a rate 29.5 faster than the moon's rate of revolution.

We always see the same side of the Moon because it takes the same amount of time (about 29.5 days) for the Moon to complete one orbit around the Earth as it does for the Moon to complete one rotation about its axis.

Now you're probably wondering why it takes the same amount of time for the Moon to complete one revolution around the Earth as it does for it to complete one rotation about its axis. This is a phenomenon known as tidal locking. The force of Earth's gravity pulling on the Moon causes it to bulge slightly in the direction of the Earth. Now imagine that the rotation of the Moon about its axis were faster (or slower) than its orbit around the Earth. Because the force of gravity is inversely proportional to the square of the distance between two bodies (F=GMm/r², where M and m are the masses of the 2 objects, G is the gravitational constant, and r is the distance between the two objects), the force on the bulge closer to the Earth is greater than the force on the bulge farther from the Earth. Thus the force of Earth's gravity pulling on the bulge would produce a torque (a force that causes rotation) on the Moon, causing the Moon's rotation to slow down (or speed up), until the period (amount of time to complete one cycle) of the rotation of the Moon about its axis and the orbit of the Moon around the Earth are the same.

I have also attached a drawing I made to help illustrate this concept.