Demolition removes the tub, the surround, and usually the wallboard behind it. The plumber then moves the drain to the shower location, sets a new trap, and installs a new valve, all under a plumbing permit filed online with the town. The inspector checks that rough plumbing and signs the permit posted at the house.
After the inspection, the remodeler installs a base or builds a tiled pan, waterproofs the walls, and finishes with tile or panels and a door or fixed glass. Whether a building permit is also needed depends on the scope, so ask the remodeler to list every permit in the estimate.
In a house from the 1960s, the subfloor under an old tub is often fine; in an older home it may be stained or soft. Either way, the remodeler should show you what they find before the new base goes in.