What does it mean when they say a song is a 'Cover'?

Usually, it's a song made famous by someone else. And of course there are cover bands. But really, the first version need not be a big hit. Covers of songs have surpassed the original one in both sales as well as artistically. "Hurt" comes to mind.

Surely, if you were doing another copy of your own song, you would call it a "Re-mix" rather than a cover???

A cover does imply that you are covering something that another artist/band has done before you.

If you are covering a song, you're performing a song that was originally sung by someone else.

If you've composed a song that contains a portion of someone else's original work, that is a "sample" or you're sampling.

It simply means someone is singing or performing a song originally written by someone else.