Fast roping is a method for rapid insertion of soldiers into a battlefield. Technically it is an air assault method where a large numbers of soldiers drop from a hovering helicopter with a large, thick, rope hanging from it. The soldiers are not connected to the rope; they just jump on and slide down, similar to firemen down a pole.
Fast rope insertion is not taught at air assault school. It is a method of insertion reserved for elite units such as Rangers or Delta Force. If the mission calls for extremely quick action soldiers will be on the rope as soon as it goes out the door. If the pilot is slightly too high, the rope won't reach the ground and may result in a bad fall for the soldier. Normally soldiers don't get on the rope until it hits ground. Fast rope operations usually take place at a height of 35 feet.
During the Battle of Mogadishu a Ranger, PFC Todd Blackburn, missed the rope and fell approximately 70 feet from a BlackHawk helicopter. He survived.
See also:
http://www.safetycenter.navy.mil/media/groundwarrior/issues/Winter01/nexttime.htm
http://www.adtdl.army.mil/cgi-bin/atdl.dll/tc/21-24/toc.htm