A corkscrew is basically a screw thread on a handle.  The origin, according to Wikipedia is "the gun worm, a device used by musket men to remove unspent charges from a musket's barrel".

The simplest design is just a crossways handle on a screw, at right angles to it.  You screw it in to the cork and pull it out, using elbow grease.  The handle may be ordinary wood, turned to fit comfortably in the hand, a novelty item such as a 303 bullet, (that should make the party go with a bang), or something elaborate like a varnished piece of vine stem.  Other designs incorporate some form of leverage against the bottle top.

There are two sorts of screw:

central shank with a flanged thread, like a screw helical wound, like a spring.

The latter is better as it spreads the load on the cork more evenly.  A stiff thin spring with a wide coil and sharp point is best.  The central shank ones can pull out the center of the cork if it is weak or crumbly, and can also drop bits of cork in to the wine if the point penetrates the bottom of the cork.  The spring type is designed to pierce right through the cork without breaking it, although this can still happen with poor corks.

There are many designs, but common ones are:

• Butterfly:  As the cork is penetrated, the arms raise.  Pulling the arms down again extracts the cork.
• Waiters Friend:  Fits in the pocket.  It can be hard to use unless of the sort with two leverage shoulders and a hinge between them, as shown.
• Screwpull:  Easy to use, and has a well designed screw.
• Butler's friend.  Suposedly so called as it allowed removal of the cork without damage, thus allowing sampling, top up and replacement.

One can get elaborate gift sets.  One of which incorporates a lever system where you just pull the lever down and up again and the cork comes out as if by magic.  Tesco used to sell one for £7.96 (later £5.97). 

There are also plenty of novelty corkscrews that can be used as gifts. The Chichester Cook Shop did a range of novelty ones from Alessi, by the Italian designer Alessandro Mendini, in the shape of a man or woman. The woman is a butterfly style, I am not sure what style the man is.  Alessi also do a rather fetching waiters friend in the shape of a parrot that took my fancy.

Another that I found amusing was "Wine on tap".

• The Virtual Corkscrew Museum - http://www.bullworks.net/virtual.htm
• The Wine Doctor - http://www.thewinedoctor.com/advisory/openservecorks.shtml
• The Corkscrew Centre - http://www.corkscrewcentre.com/