The right pedal is the sustain pedal (damper pedal), it raises all of the dampers at once, allowing all notes which are played to continue sounding after the keys are released.
The left pedal is the soft pedal. In most grand pianos, depressing the pedal shifts the action to the right slightly. This allows the hammer to strike only two of the three strings in its unison, thus reducing the total volume. This type of pedal is called the una corda pedal. In vertical pianos, and some grands, depressing the pedal shifts the hammers closer to the strings, thus decreasing their traveling distance and the total force with which they can strike the strings.
The middle pedal is called the sostenuto pedal. This pedal is found only in better grand pianos. Depressing this pedal will sustain only notes which are already depressed, freeing your hands to play any other notes. However, notes played after depressing this pedal will not be sustained by it. Many vertical pianos, and some grands, delegate various other functions to this middle pedal. It can be used as a “practice pedal”, muting the overall volume of the instrument, or as a sustain pedal for just the bass area of the piano, or as an effects pedal, lowering a strip of metal studs between the hammers and strings, creating a honky-tonk type of sound.