Having the sound precede the visual is called a J cut; having the sound (from the previous scene) follow the visual change is called an L cut. This is from the visual appearance of the editing software, where video is in the top half of the sequence editor and audio in the bottom. The shape of the protruding audio clip relative to the vertical video cut loosely resembles the letters J and L.

It's called a J-cut: http://movies.stackexchange.com/questions/18690/sound-of-a-n...

The technique when talking about screenwriting or film analysis is a prelap. It's the opposite of a bridge, in which the sound from a previous scene carries into the next.

Ostensibly, this is a reflexive design element, though these things can have sublte connotative/implied ideas. Sometimes the prelab audio matches both pre- and post-cut scenes, and help you to think about the content in a different way. Or, it may just be an interesting way to cut your content. :-)