Tidal locking makes one side of an astronomical body always face another, like the Moon and the Earth. A tidally locked body takes just as long to rotate around its own axis as it does to revolve around its partner. In the case of the moon, this period is just over 4 weeks, and no matter where you are on the earth you always see the same face of the moon. (The other side of the moon was not glimpsed until the 1960s, when photos were taken from spacecraft).
The condition occurs in astronomical bodies such as planets and moonss that orbit each other closely. It results in the orbiting bodies synchronizing their rotation so that one side always faces its partner (or, alternately, places them in a resonance). Tidal locking can potentially occur in any orbiting object, not just planetary ones.
Gravitational attraction between two bodies produces a tidal force on each of them, stretching each body along the axis oriented towards its partner and compressing it along the other two perpendicular axes. If the bodies in question have sufficient flexibility and the tidal force is sufficiently strong, this will distort the orbiting bodies' shapes slightly. Since most moons and all larger astronomical bodies are roughly spherical due to self-gravitation, this causes them to become slightly prolate (egg-shaped).
If either of the two orbiting bodies is rotating relative to the other, this prolate shape is not stable. The rotation of the body will cause the long axis to move out of alignment with the other object, and the tidal force will have to reshape it to restore the situation. In a sense, the tidal bulges "move" around the body as it rotates to stay in alignment with the body producing it. This is most clearly seen on Earth by how the ocean tides rise and fall with the rising and setting of its Moon, but it occurs on all rotating orbiting bodies.
The rotation of the tidal bulge out of alignment with the body that caused it results in a small but significant force acting to slow the rotation of the first body relative to the second. Since it takes a small but nonzero amount of time for the bulge to shift position, the tidal bulge of the satellite is always located slightly away from the nearest point to its primary in the direction of the satellite's rotation. This bulge is pulled on by the primary's gravity, resulting in a slight force pulling the surface of the satellite in the opposite direction of its rotation. The rotation of the satellite slowly decreases, with its orbital momentum being boosted in the process. Note that this assumes that the satellite is rotating more quickly than it is orbiting its primary. If the opposite is true, tidal forces increase its rate of rotation at the expense of orbital momentum instead.
Almost all moons in the solar system are tidally locked with their primaries, since they orbit very closely and tidal force increases rapidly with decreasing distance. In addition, Mercury is tidally locked with the Sun in a 3:2 resonance. More subtly, the planet Venus is tidally locked with the planet Earth, so that whenever the two are at their closest approach to each other in their orbits Venus always has the same face towards Earth (the tidal forces involved in this lock are extremely small and it may be primarily a result of coincidence; see Venus' article for more detail). In general, any object that orbits another massive object closely for long periods is likely to be tidally locked to it. Close binary stars throughout the universe are expected to be tidally locked with each other, and extrasolar planets that have been found to orbit their primaries extremely closely are also thought to be tidally locked to them.