There are several ways of severing the connection between the glider and the winch.
Firstly, at the glider, there is the connector that can be either opened manually by the pilot (by pulling the yellow lever/ball/whatever the aircraft has) or will release the ring by itself when pulled back under a certain angle. After the disconnect, the pilot will actuate the release lever a few times by himself to ensure no cable left connected. These are the normal ways to end a winch launch.
Secondly, the cable might break either at the weak link, the parachute or some point in the cable itself. This requires the pilot to fly the above mentioned procedure, not forgetting to drop whatever rope may be still attached to the aircraft again by pulling the release, the ground crew to search for the loose cable and the winch driver to repair the rope using one of the many ways available (occasionally a bloody business).
And thirdly, if the winch driver sees the glider pass behind the winch without dropping the rope or if he is otherwise unhappy with the gliders behaviour (e. g. if it underflies the cables parachute or makes weird movements while under tow), he can use the "guillotine" to cut the cable at the winch. This one typically consists of a blade mounted on a pre-tensioned strong spring that when released will cut through the cable and is required to show its ability to cut several strands of tow line (IIRC, 3 or 5 of them - it has been a while since I last operated a glider winch) at once during the annual check.
This of course leaves the glider with several 100m of cable attached. The options are generally the same as with a breakage plus the silk way down in case of sufficient height: having the line entangled somewhere on the ground during approach will easily lead to nasty consequences, so this option may well be considered as well.
I heartily agree with the words on safety around a winch. A parachute with the weak links to the head can ruin ones afternoon quite well, as can receiving a hit from a broken line. Once after a rather late disconnect of the glider (IIRC, a slight tailwind was involved), I had the parachute assembly come down right beside the winch and hit the roof of the old vehicle used to return the lines to the launch side. The driver was rather white about the gills when he got out, but fortunately received no injuries. He made sure to park a bit further behind the winch after that.