Screws are amazingly versatile and powerful fasteners. The basic concept is used for holding all sorts of things together including wood, plastic and metal. Different varieties have been developed to maximize efficacy for particular applications.

Driving Methods

Slotted: Slotted screw heads are perhaps the oldest and most common variety; a linear slot in the head accepts a standard (flat) screwdriver.
Phillips: The phillips head is an improvement to the slotted screw; cross-shaped grooves accept a phillips-head screwdriver. These heads provide a larger mating surface between the screw and the screwdriver which minimizes wear and helps prevent slipping.
Hex: These heads do not have any grooves or slots at all; the entire head is hexagonally shaped and they are tightened with a wrench. Since the size of the heads varies, a wrench set or an adjustable wrench is required for driving hex screws.
Square: (also know as Robertson drive). These heads have a square indentation to minimize slipping; they also require a special driver for tightening and loosening.
One Way: One-way heads are a variation on the slotted screw. They can be tightened with a standard screwdriver, but are tamper-resistant because they require special tools for removal.
Torx: Torx heads have 6-pointed recessions to provide even more surface area for driving. They require the less-common torx screwdrivers and therefore can be considered tamper-resistant.
The various driving methods listed above can be combined with various head shapes below. For example, you can have a phillips flat head screw or a phillips button head screw.

Head Shape

Pan Head: Standard screw head profile with average diameter and average height.
Button Head: Similar to a pan head but with a curved top.
Round Head: More pronounced head than a button head.
Flat Head: Flat top with a tapered underside that is intended to be driven into a countersunk hole.
Oval Head: Tapered underside like a flat-head, but with an oval top.
Truss Head: These are large-diameter heads with a low profile.
Fillister Head: Thick profile with a slightly rounded top.
In addition to all the head options that are described above, there is a large variety of screw bodies:

Thread Varieties

Wood Screw: These screws are used in wood. The threads are usually coarse and deep to help them grab the wood.
Machine Screw: These screws have finer threads than wood screws. They are designed to be used in conjunction with a nut or tapped hole.
Sheet Metal Screw: These screws are usually short and have coarse threads designed to grab onto relatively thin sheet metal.
High-Low: These screws have two sets of threads with alternating heights. High-low screws are specifically designed for certain plastics and other low-density materials.
Self-Tapping: Self-tapping or thread-forming screws feature threads designed to tap their own holes. These work well in softer materials such as wood and plastic but are not suitable for harder materials.

In addition to all the varieties described above, many screws are made out of different materials. They can be made out of stainless steel, steel, brass, aluminum, teflon, nylon etc. Screws can also have various finishes such as zinc plating, black oxide or non-stick coating; screws that will be exposed to the elements may be galvanized to prevent rusting. Needless to say, there are thousands and thousands of options when you consider that screws come in different diameters and lengths!