A screw is a threaded fastener whose major diameter, which is to say its outer thread size, is less than one-quarter inch (SAE) or for metric screws, 6 millimeters. Anything that size or larger is a bolt. At the top they feature a head which serves both to hold down material (their primary purpose being to fasten two or more items together)
Screws are offered in any size (below 1/4" that is), pitch, and material, and with almost any kind of head you can imagine - there are round headed screws with indentations for flat-blade, philips, torx, and other types of screwdriver, cap headed screws which usually have hex or torx depressions, hex-headed screws which look like miniature bolts, taper headed screws which are designed to screw down into conical depressions and lie flush with the surface they pass through, and more.
While SAE bolts are measured by their major diameter in inches or fractions thereof, SAE screws are based on several screw sizes, measure in numbers from zero to twelve, excluding seven, nine, and eleven. Each screw also has a corresponding fine thread pitch, and a coarse thread pitch measured in threads per inch - except for #0, which has only a fine pitch. These threads are known as "national fine" and "national coarse" thread pitches.
0 - .060 in.,
80 t.p.i. fine
1 - .073 in.,
72 t.p.i. fine, 64 t.p.i coarse
2 - .086 in.,
64 t.p.i. fine, 56 t.p.i coarse
3 - .099 in.,
56 t.p.i. fine, 48 t.p.i coarse
4 - .112 in.,
48 t.p.i. fine, 40 t.p.i coarse
5 - .125 in. (1/8"),
44 t.p.i. fine, 40 t.p.i coarse
6 - .138 in.,
40t.p.i. fine, 32t.p.i coarse
7 - No such size
8 - .164 in.,
36t.p.i. fine, 32 t.p.i coarse
9 - No such size
10 - .190 in.,
32 t.p.i. fine, 24 t.p.i coarse
11 - No such size
12 - .216 in.,
28 t.p.i. fine, 24 t.p.i coarse
Metric screws are measured more simply, and their sizes specified with three numbers; major diameter in mm, distance between threads in mm, and length in mm. Therefore a 5mm screw with a thread every 0.8mm (standard metric pitch for that size) two centimeters long would be shown as M5x0.8x20, with the M naturally denoting "metric".
You typically will not see a screw smaller than 4mm, which has a 0.7mm space between threads. 6mm screws ordinarily have 1mm between threads.
These sizes only really apply to machine screws however. A wood screw will have a thin thread with more space than thread. Where the american standard thread has a 60 degree thread form with as little flat area between threads as possible, wood screws maximize flat area due simply to the nature of wood. They tend to have few threads per inch. Not all wood screws have the same pitch, either; A wood screw meant for applying a significant amount of force might have as many as 12 or 14 threads per inch, while a screw meant for decking might have as few as six. Drywall screws are similar to wood screws, but typically have a smoother finish and a head which has a taper in the shape of an arc to avoid cracking the plaster.
Screws are a combination of two simple machines, the wheel and the inclined plane. An inclined plane is a lever with an infinite fulcrum (the fulcrum corresponds with the surface of the plane) and a wheel is an infinite series of levers rotating around a central fulcrum, thus the screw can be seen as an assortment of levers. Even the force applied to a screw is measured in torque, which is a measurement of force over distance, using a lever.