First, look closely. Look around the edge of the back for a small lip or fingernail notch, which usually means a hinged or snap back. Fine grip ridges, or a plain back with no hinge visible, usually mean a screw back. Also check whether the watch is a hunter, with a hinged front cover that opens when you press the crown.
Hinged snap backs: hold the watch face-down in a soft cloth. Put a thumbnail, or a proper case knife held flat, into the lip opposite the hinge and lift gently. There is often an inner dust cover (a 'cuvette') under the outer back. It opens the same way, and the movement is visible under it. Never lever against the hinge side.
Screw backs, common on American railway and later watches: grip the back with a rubber ball or rubber jar-opener pad and turn it anticlockwise, while holding the bezel with your other hand. If it does not move with moderate force, stop. Old grease and corrosion can lock the threads, and forcing them can distort the case.
Snap-on and screw bezels: some watches are reached from the front. The crystal bezel either snaps off or unscrews, and then the movement swings out on a hinge or comes out through the front. Treat the crystal gently, because old glass and acrylic crack easily.
Key-wind watches: the key-winding and setting arbors are usually under the inner cover or, for setting, under the front bezel. Open the cover to reach them. Do not push a key onto a square it does not fit.
What not to do: do not use kitchen knives, screwdrivers or blades that can slip into the movement. Do not breathe onto, touch or blow into the open movement. Do not touch the balance wheel or hairspring. If the back is dented, stuck or has no obvious way in, take it to a watchmaker. A slipped tool can do more damage in a second than decades of use.