What: Dice is a culinary term. As a noun, it refers (ideally) to cubes of chopped foodstuff — usually vegetable, between ¼ and ¾ inches on a side. As a verb, 'to dice' refers to the process of preparing a dice — making the cuts that lead to those little cubes of food. Contrast a dice with a chop (larger and less regular) and a mince (much smaller irregular cuts).
Why: The decision to chop, dice or mince may at first be considered trivial, but this is not so. The most obvious consideration is the treatment of the vegetables. A light treatment — whether cooking, saucing, preserving or whatever, calls for a smaller cut so that more surface area is exposed to the treatment. A more heavy handed treatment of the vegetable calls for a larger cut, such as a chop. Also, you must consider the opposite relationship. The flavor of your vegetable will spread to surrounding ingredients through those cut surfaces. E.g. You often mince garlic because the point is to permeate the dish with the garlic's flavor. The final primary consideration is cooking time. When cooking e.g. chard or broccoli the stems of the plant have a distinctly different size and density than the leaf and flower. A small dice of the stems may make both parts cook to completion in the same time as the outermost parts.
How: The basic dicing instructions that you find across the sources sort of assume a round vegetable — an onion is a common subject. Some places suggest that you should be cutting off the curved edges so that you can work with a rectangular solid — promoting more perfect cubes, but most of the sources are less extreme. In general, cut round things (potatoes, onions, apples, whatever) in half first and lay the newly created flat side down on your cutting board. From here, you slice the vegetable into panels the width of your selected dice. At this point, you're going to cut the panels into batons (I like batons (from Joy of Cooking) better than matchsticks, strips or sticks). How you do this depends on what you're cutting and how regular it is. In the ideal, you have perfectly rectangular panels and you just stack them up and slice through several panels at a time to form perfect batons. Professional onion cuttery has you pressing the root of the almost halved onion down to the cutting board and making your slices parallel to the surface of the cutting board. If you're dicing e.g. potatoes, and not discarding the rounded edges, you have to deal with the natural curves of the vegetable. Stacking is still your best bet, but more attention is required to prevent small slivers and large chunks from entering your dice.
Variations: Obviously most things aren't square, but some things that you might want to dice, aren't even round. Like celery. Or cucumbers. To dice a cuke, cut it into two-inch sections and then follow the procedure above. To dice celery, if it's a large dice you really just slice it into chunks, but more commonly you'll slice each stalk lengthwise — splitting it in two and then dice those with a simple chop. Given the general rules and these two variations, you should be able to extrapolate how to dice pretty much anything that you might need to.
The Joy of Cooking, ©1997, Irma S. Rombauer, Marion Rombauer Becker and Ethan Becker.