What is a music promo?

This is an example of a promo video from Beyoncé's
 'Single Ladies'

A music promo is the copy of an album that is used for promotional purposes such as sending to the press and the radio stations/music channels before the release of the album.
This is so that they get reviews and so that the promo video/song can be played on TV and radio.
They are also sent to agents and promoters when booking the shows.

Basically a music promo is the main part of the promotion as it gets press coverage and can get record label attention. 

What is the purpose/function of a promo?

The purpose/function of promo is to get press coverage, attract record labels, attract an audience and make sure that the album, when it finally comes out, will get sales because the audience has heard a few songs and liked them enough to get the album.