On a grand piano the soft pedal, that is the one on the left, is called the “Una Corda”. This is slightly misleading as it shifts the keyboard to one side, usually to the right, so that the hammers hit only two of the three strings in the treble (that’s the misleading bit, it doesn’t strike just “one string”) or one of the two strings in the tenor section. The bottom octave or so has only one string so is largely unaffected. The result of this sideways shift is that fewer strings are struck and the volume is accordingly less. Where the hammers have been worn (the nose develops grooves from the strings), the shift also presents a new surface to hit the strings and can have a marked tonal effect as well as a drop in volume. This can be overcome by periodically refacing the hammers, that is, reshaping them by filing away the outer surface.
Uprights are quite different. There are two different types of soft pedal; the half-blow and the celeste. If you are lucky you will have a celeste pedal on your upright which, when it is set up correctly, lifts a rail with a strip of felt attached to sit between the hammers and strings thus muting the sound produced. Also known as the Good Neighbour pedal, this will help stop next door complaining after a late night session of blues improvisations, although it won’t completely negate the effects of positioning the piano against an adjoining wall.
The half-blow, on the other hand, has no known effect on the volume of the piano at all. It moves all the hammers half way to the strings so that, theoretically at least, they pick up less momentum on their short journey and strike the strings with a little less force. It also introduces lost motion into the keys which makes them feel less positive. I have often been informed that the soft pedal isn’t working but I can usually demonstrate that it is. The half-blow pedal is there for two reasons; aesthetically, it is far more pleasing to see two pedals rather than just the one sustain pedal, and psychologically, pressing the soft pedal makes one play quieter.
Sometimes, and on most modern pianos, you will see three pedals on the upright. This gives the best of both worlds as the centre pedal is the celeste and the left is the half-blow pedal.