Nathan is right. A supporting band or a back-up band is one that is not as famous as the the main band, and they are "billed" together for the same musical event. These are the less well known bands that play first to warm things up, you know, as people are still finding their seats and having their first drink. And "backing" usually means that the lesser known musicians will join and play along with the famous ones for some or all of the songs. Deep Purple and Black Sabbath are hugely famous, and I have never heard about the Tucky Buzzards!

In this context, a supporting band and a backing band are two different things. Supporting bands play before a more famous act, providing a warm-up performance for the audience. A backup band actually plays with a (usually) more famous musician such as a singer. An example of a backup band is The Heartbreakers, who played backup instruments for the singer Tom Petty.