This question is easy to answer.
It depends!
Prior to the incorporation of a middle pedal, most every piano, grand or upright, had two pedals. The one on the left made the piano play more quietly, and the right pedal sustained any notes played until the pedal was released. These outer two pedals retain the same functions in modern times.
Starting in the middle 19th century, manufacturers began to incorporate a middle pedal into grand pianos. This pedal is called the sostenuto pedal, so long as it functions as a true sostenuto. This pedal sustains the sound coming from all keys which are already depressed when the pedal is engaged. Other keys continue to play normally. You might think of this as a selective sustain pedal. Here is a brief demonstration:
Most grand pianos have a proper sostenuto pedal, but some inexpensive models do not. Almost no upright pianos have a proper sostenuto pedal, but a handful do, most notably those made by Steinway.
In any cases where the middle pedal does not selectively sustain particular notes which have been depressed, the “middle pedal” should be called exactly that, and not a “sostenuto pedal”.
Not long after the sostenuto pedal was introduced, those buying upright pianos wanted a middle pedal too, and piano manufacturers were quick to jump into satisfying that perceived need.
As mentioned, few uprights have true sostenuto pedals, so what does the middle pedal do on a vertical piano? On earlier pianos, from around the first half of the 20th century or so, better quality pianos took the function of the right (sustain) pedal and gave it to the bass section only of the piano.
In the following video, I show the sustain pedal twice, followed by the middle pedal:
And here in action, you can see that it does approximate the function of a sostenuto:
I have seen a handful of uprights where the middle pedal worked just like in the video, but the sustain pedal worked only the middle/treble section, effectively dividing the sustain pedal in half. On those pianos, to get the function of a regular sustain pedal, you have to step on both the middle and right pedal at the same time. Not the best design.
On quite a few lower quality uprights, the middle pedal is tied to the right pedal, so they both do exactly the same thing – they both behave as a standard sustain pedal. This demonstrates how well advertising sometimes works on the masses – the middle pedal on these pianos exists merely so that there is a 3rd pedal.
In modern times (the last 50 or so years), we most often find that the middle pedal on vertical pianos lowers a mute rail (muffler/practice rail) in between the hammers and strings. This long strip of felt quiets the sound of the piano considerably. Some manufacturers have added a notch in the travel of the middle pedal which allows the player to push the pedal sideways into the notch, locking it into position. This frees up the left foot to work the left pedal.
Here is what a mute rail looks like as it drops into position:
These are just the most common ways that the middle pedal has been implemented into the piano world, no doubt there are others. I hope this has been helpful in answering this oft asked question.
-Joe, from Burd and Sons Piano Service