Most rolltop desk covers use a heavy-duty fabric as backing. The fabric must be strong enough to withstand the pressure as the desk cover opens and closes.
Most rolltop desk covers are made from canvas, although you can use other fabrics as long as they are heavy enough to stay together when you use the rolltop desk cover.
The fabric is held to the back of the desk cover with fabric glue. This glue holds the canvas against the cover's wooden slats so that it moves with the slats.
When you replace the backing, you don't remove the old backing — you glue the new backing to the old backing with fabric glue so that the backing no longer separates from the wooden slats of the desk.
Some rolltop desks don't use any kind of fabric backing. Instead, these desks use wooden slats to make the cover and ball-and-socket joints to support the wooden slats.
If a slat breaks, obviously you will need to replace the wood. Otherwise, you don't need any kind of backing material on this type of rolltop desk.
If your rolltop desk does use a cloth backing, you'll need materials to install the new backing. Most fabric glue needs to sit in a plastic foam bowl to set adequately, and you'll need a paintbrush to apply the fabric glue to the fabric.
Most of the time, it's not a good idea to remove the old fabric, but if the fabric is already largely off, you can use vinegar to loosen the old glue and remove the old fabric.