All the answers so far are certainly descriptive of the winglets, however, having been an airline pilot for many years, I don't see the basic science well explained.
Firstly, like a lot of other inventions, the winglet phenomena actually came from the light aircraft sector. Burt Rutan, in his homebuilt
Vari-eze was probably the first inventor and Peter Masak thought the process through for gliders and it was later adapted by several
homebuilt. adaptations.
Secondly, in flight, because of the design of the wing, a higher air pressure is developed on the underside of the wing as compared to the pressure on top of the wing. At the end of the wing (wing-tip) the high presure air tends to spill off into the lower pressure area creating a great deal of turbulence. the phenomena is called (as related in some of the previous answers) as wing tip vortex. Engineers have tried to deal with this energy robbing, efficiency wasting fact for many years. In fact, one of the contributors to higher efficiency was the development of the
high aspect ratio wing which dealt with some of the spillage. The winglet, as the charts, graphs and depictions illustrate lessens the effects greatly...
Some winglets don't look as dramatic as the ones discussed here.
Airbus has a device which is actually fastened to the wing tip that provides, to a lesser degree, the same type of benefit... seen here:
http://www.airliners.net/open.file?id=0969902. .. Many other variants exist or are in further development...