20/x (or 20/20, or 20/400) is called "visual acuity" and gives a descriptive measurement of your vision. It basically means that at 20 feet, the smallest letter you can see on the chart could be seen by a person with "normal" vision at a distance of (the second number) feet. Therefore, 20/20 is considered "normal." 20/200 would mean that your vision is the same at 20 feet as a "normal" person's vision is at 200 feet.
The diopter is a measure of the strength of a lens. It is the reciprocal of the focal length of the lens (d = 1/f).
A plain lens is a diopter of 0. Positive diopters are for magnifying lenses, and correct farsightedness. Negative diopters are for reducing lenses, and correct nearsightedness.
In addition to diopter, glasses may also have several other corrections not included with most contact lenses - cylinder, angle, and add-power. The cylinder and angle correct astigmatism, and add-power (bifocal, multi-focal) corrects for presbyopia, the thing us old folks get...