To upholster is to cover a piece of furniture with fabric and soft padding. If your brother's cat scratches up your sofa, you may want to upholster it — but it's probably wise to wait until your brother and Mr. Whiskers move out.
Upholstery is everything that covers your couches, love seats, and soft chairs, from the springs to the velvet cushions. The act of making these furniture coverings is also upholstery, and when you do it, you upholster. The word for the expert who specializes in this, upholsterer, came first, from the Middle English upholden, "to repair."