Picture drilling a hole 4000 meters vertically and then turning that hole 90 degrees to drill another 1000 meters horizontally. That is horizontal drilling.
Drilling horizontally allows access to gas that might not be accessible by drilling straight down. Gas under lakes, parks, railroad crossings, environmentally sensitive areas, etc are either too difficult or cause too much damage if the drilling is done directly above the formation. Another reason to drill a horizontal section is so the pipe occupies a larger area of the formation to recover more gas.
This is a rough idea of the concept. At the 1:30 minute mark, the grey substance being injected is cement. The cement covers the outside of the case tubing and prevents the hole from caving. It also keeps the oil from contaminating anything above the zone in which it is trapped as well as preventing ground water from entering the production zone. The only escape for the oil or gas becomes the pipe that is inserted inside the tubing (seen towards the end of the video).
I took this video on October 18, 2008. It shows a drilling crew dropping pipe back into the hole after changing a drill bit. The pipe you see is the pipe that will bend to the horizontal position. The bending takes place at 8 degree increments for every 30 meter drop taking about 330 meters of vertically depth to complete the turn.