It is my attempt to keep these answers as non technical as possible, so they can be understood by all. Please forgive me if I fail :-P

A halftone originally was a symmetrical grid of dots ( Called a screen) used to reproduce a continuos tone image like the Classic t-shirt photograph of Brando in illustration one.

Over time it has come to also mean a percentage of colour reproduction, such as half of a tone (See dictionary.com) This was originally called a screen tint and is illustrated by the Colour squares in Illustration two.

Halftones are measured by the number of dots in a given inch or the lines between each dot in an inch. this always equals 100% so if there are ten dots per inch the lines will be 90, or 40 dots and 60 lines, 50 lines-50 dots.

Halftones come in many shapes like round, square, oval, diamond, lines and even custom shapes like a company logo.

With today's Digital equipment, Halftones can range from just a few dots per inch to many thousands of dots. the smallest dots cannot even be seen by the human eye.

Keep in mind, due to the Grid of dots in your computer monitor, depending on your resolution settings you may be seeing Moire' patters in my illustrations. see FAQ on Moire'

Read a more in-depth Geek version @ wikipedia