Halftone images are surprisingly simple in practice. Originally, they were designed to create images that only required one color of ink (usually black), but still needed to offer a good range of tones. For example, black and white photography consists of only one color of ink: true black. This means that when viewed under a microscope, there will only be two colors present in any black and white picture: black and white. The range of different grays we see is actually produced by varying the size of individual dots of ink, and these dots are aptly named halftone dots.
Halftone dots are small dots of ink -- sometimes circular, elliptical, or even square -- that are spaced evenly across areas of a black and white image. By changing the size of the dot, you change the saturation of ink, and therefore can lighten or darken the tones as needed. Smaller dots produce lighter grays and off-whites, while larger dots can lead to dark gray or black areas in an image.
Not only does this save on ink and reduce the number of expensive colors of ink, it's actually interesting from an optical illusion standpoint. Because the human eye can only see tiny dots up close, halftone images from a distance appear to have a rich depth of tones from true white to true black, as well as many shades of gray in between. Like most things, the original halftone images were of lower quality, but thanks to advancements in technology, halftone images can provide a rich viewing experience. Printers today actually use the same concept when it comes to printing in colors. By using the CYMK color palette which consists of only four inks - cyan, yellow, magenta, and key (black) - printers can layer together different colors of dots to produce millions of colors and a rich, crisp image.