The Basic Corkscrew
A basic corkscrew features a handle with a spiraling, screw-shaped wire. The wire is placed against the cork and driven into it by turning. Every undulation of the screw allows it to grip a different part of the cork, greatly increasing the surface area pulled on, while also distributing the force across the cork. Past that, the basic corkscrew works through the simply muscle power of getting a good grip on the bottle and pulling.
This corkscrew is similar to the basic corkscrew, with the screw being driven into the cork and increasing the surface area for drawing the cork, but with the added element of having a hinged lever. This lever has a notch for bracing against the lip of the bottle, and adds leverage, making pulling the cork out much easier. They also usually come with a small knife for removing bottle foil.
This device usually combines a corkscrew with a bottle cap opener, which replaced a proper handle and therefore can make turning the screw a little difficult. The screw is driven into the cork as usual, until the head of the corkscrew is in contact with the lip of the bottle. Then the wings (levers on the side of the corkscrew) are pushed down, and a rack and pinion system adds mechanical advantage and makes drawing the cork much easier.
This is a somewhat antiquated device, which was also called "the butler's friend." It is the only corkscrew that does not feature an actual screw. Instead, the handle has two prongs, which are pushed into the bottle along the neck, pinching the cork away from the neck. The cork is then twisted out. This is difficult to use properly, but it does have the advantage of drawing a cork without damaging it.