Today, we are going to talk about the function of the middle pedal in a grand piano.

The pianos were firstly designed with only 2 pedals, with the left always being some types of soft pedal and the right being the sustaining pedal. The middle pedal is added lastly in the historical development of pedals to the pianos. 
In most cases, the middle pedal on a grand piano is the sostenuto pedal because this pedal also sustains the notes of keys.

Confusing, right? Because the right pedal is the sustaining pedal.....

Here is the difference:
Sustaining pedal (the right one) sustains EVERY note that is played as long as the pedal remains pushed down.
Sostenuto pedal (the middle one) keeps ONLY the notes whose keys you hold down PRIOR TO and WHILE the pedal is pushed down, any note that comes after the sostuneto pedal being pushed down is not sustained.

Perhaps an example is necessary for explanation:
To see the difference of functionality of the pedals, let's play 3 notes (C,D &E) in a consecutive manner:

Depress the sustaining pedal, followed by playing the 3 notes consecutively:
Result: You hear all 3 notes were sustained as long as the sustaining pedal remains pushed down.

Press down the sostuneto pedal, followed by playing the 3 notes consecutively:
Result: You hear distinct C,D &E notes consecutively, non note is sustained.

But if you
Hold down the key for note C, depress the sostuneto pedal, release the key for note C, continue to press keys for note D and E while the sostuneto pedal remains depressed:
Result: Note C is sustained, but not note D nor E
Reason: When the sostuneto pedal is depressed, the damper for note C strings is 'already' lifted away from the strings, press down the sostuneto pedal 'at this moment' keeps this particular damper remains in the lifted away from the strings position, but same effect does not apply to note D and E because these 2 notes were played after this pedal has been pushed down.

This is what the sostuneto pedal does in most of the grand pianos, sustaining only selective notes instead of all the notes.
However, in very few types of grand pianos, the middle pedal is the bass sustaining pedal, not a sostuneto pedal. The bass sustaining pedal lifts only the bass dampers. As a result, only the bass notes are sustained while the pedal is depressed (like a sustaining pedal, but only for the bass notes).