The headliner used to be a lightweight fabric that was reinforced with some ribbing and glued directly to the inside of the roof, a method of installation that almost always fails in time. Fabric headliners are prone to tearing, discoloration and sagging as they age.
As technology has evolved, so has the headliner. First, manufacturers experimented with adding a layer of insulation beneath the fabric, which worked pretty well, but still left the headliner vulnerable to the ravages of wear and tear and sagging.
Eventually, the headliner was bonded to cardboard or plywood backing, which was then securely fastened to the roof, which lasted longer than loose fabric.
Today, manufacturers are leaning toward molded, one-piece headliners which fit precisely into the body and include recesses for lights, visors and storage compartments.
You can still find new cars with fabric headliners -- in fact, a popular luxury upgrade is to replace the fabric headliner with a leather or brushed suede one. It's a swanky look and feel that will age nicely, and outlast most fabrics.

