Why do we only ever see the same side of the moon?
If this is to do with gravity are there any variables which mean we might one day see more than we have before?
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The reason for this is what we call tidal locking:
Tidal locking (or captured rotation) occurs when the gravitational gradient makes one side of an astronomical body always face another, an effect known as synchronous rotation. For example, the same side of the Earth's Moon always faces the Earth. A tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner. This causes one hemisphere constantly to face the partner body. Usually, at any given time only the satellite is tidally locked around the larger body, but if the difference in mass between the two bodies and their physical separation is small, each may be tidally locked to the other, as is the case between Pluto and Charon. This effect is employed to stabilize some artificial satellites.
Fig. 1: Tidal locking results in the Moon rotating about its axis in about the same time it takes to orbit the Earth. (Source: Wikipedia)
Fig. 1, cont.: Except for libration effects, this results in the Moon keeping the same face turned towards the Earth, as seen in the figure on the left. (The Moon is shown in polar view, and is not drawn to scale.) If the Moon were not spinning at all, it would alternately show its near and far sides to the Earth while moving around our planet in orbit, as shown in the figure on the right.
Fig. 2: Lunar librations in latitude and longitude over a period of one month (Source: Wikipedia)
Libration is manifested as a slow rocking back and forth of the Moon as viewed from Earth, permitting an observer to see slightly different halves of the surface at different times.
There are three types of lunar libration:
Libration in longitude results from the eccentricity of the Moon's orbit around Earth; the Moon's rotation sometimes leads and sometimes lags its orbital position.
Libration in latitude results from a slight inclination between the Moon's axis of rotation and the normal to the plane of its orbit around Earth. Its origin is analogous to how the seasons arise from Earth's revolution about the Sun.
Diurnal libration is a small daily oscillation due to the Earth's rotation, which carries an observer first to one side and then to the other side of the straight line joining Earth's and the Moon's centers, allowing the observer to look first around one side of the Moon and then around the other—because the observer is on the surface of the Earth, not at its center.
The period of rotation of the moon is ~27.322 days, and the period of revolution is also ~27.322. This means that for every degree it turns around the Earth, it turns a degree around itself, so the same side always faces us.
This is due to tidal forces coupling the various oscillators in the system (revolution of moon, orbit of moon, revolution of Earth). When oscillators are coupled, they have a tendency to settle to a state that is either in phase or 180 degrees out of phase*. Both cases give rise to tidal locking here.
As an complement to the other answers, let me address the question of why planets tend towards tidal locking. In short, the torque applied by the differential gravitational force between both sides of the surface of the planet induces friction, which in turn dissipates aways the excess spin of the (proto) moon when it is not tidally locked. When locking occurs the dissipation is minimized.
Another artifact is that the moon is also moving away from the earth (as it looses angular momentum). See e.g. http://curious.astro.cornell.edu/question.php?number=124
The other answers here are fantastic at explaining in a technical sense.
For an everyday example, imagine taking something asymmetrical--like a marble with some clay stuck to it--and spinning it. The asymmetry eventually brings the object to spin in a certain way. The moon is like that, except way more complicated, since its interaction with Earth is a part of the equation.
RE: Always seeing the same side of the moon. It may also be attributed to a difference in density of the moon. It was established recently that there are gravitational variations around the moon that would support the possibility that there is a higher density of material on one side of the moon. The higher density area of moon would then always face the earth. This would be similar to taping a heavy weight to one side of a ball on a hard surface. The heavier side of the ball will always face down towards the floor and towards the gravitational force. This is the same result of the heavier side of the moon facing towards the gravitational force emanating from the earth. It's too much of a coincidence that the moon rotates perfectly to always keep the same side facing towards the earth. Especially since the moon is moving away from the earth and is changing it's distance and time to revolve around the earth constantly. I came up with this theory about 10 years ago and I have not seen any other theories that would convince me that this is not a real possibility. This theory would also explain why Mercury always faces the same side towards the sun. There are no tides on Mercury to affect how it is orientated and it would appear that both bodies are affected by the same forces in keeping the same sides facing the same direction.