A sugar-phosphate backbone is one of the main components of DNA and RNA. It consists of 5-carbon molecules which are bonded to phosphate groups via a phosphodiester bond. This bond is on carbon 4 of the molecule and bonds through a CH2 group attached to the phosphate. These interactions continue all the way up the DNA molecule. The nucleotide bases attaching to another carbon on the carbon molecule and stick out to the side which gives the illusion of the sugar-phosphate links being a 'backbone'.
One main importance for them in DNA function is when a double helix is formed, the backbone provides some protection for the bases that have joined up.