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The sustain pedal (or, damper pedal) is often simply called "the pedal", since it is the most frequently used. It is placed as the rightmost pedal in the group. It lifts the dampers from all keys, sustaining all played notes. In addition, it alters the overall tone by allowing all strings, even the ones not directly played, to reverberate.
The soft pedal or una corda pedal is placed leftmost in the row of pedals. In grand pianos it shifts the entire action/keyboard assembly to the right (a very few instruments have shifted left) so that the hammers hit two of the three strings for each note. In the earliest pianos whose unisons were bichords rather than trichords, the action shifted so that hammers hit a single string, hence the name una corda, or 'one string'. The effect is to soften the note as well as change the tone. In uprights this action is not possible; instead the pedal moves the hammers closer to the strings, allowing the hammers to strike with less kinetic energy. This produces a slightly softer sound, but no change in timbre.
On grand pianos, the middle pedal is a sostenuto pedal. This pedal keeps raised any damper already raised at the moment the pedal is depressed. This makes it possible to sustain selected notes (by depressing the sostenuto pedal before those notes are released) while the player's hands are free to play additional notes (which aren't sustained). This can be useful for musical passages with pedal points and other otherwise tricky or impossible situations.
On many upright pianos, there is a middle pedal called the "practice" or celeste pedal. This drops a piece of felt between the hammers and strings, greatly muting the sounds. Often this pedal can be shifted while depressed, into a "locking" position.