Chopping, mincing and dicing are all cutting techniques. Chopped pieces of a given food are larger than diced pieces, and minced pieces are smaller and more irregularly shaped than diced pieces.
When you dice a food, you are cutting it into small, even pieces. Diced foods are usually cut to between 1/8 to 1/4 inch in size. A coarse dice calls for larger cubes, a medium dice for medium-sized cubes and a fine dice for small cubes.
There is not a standard size for chopped pieces to be, but in general they are usually bite-sized. As with dicing, the same size terminologies apply—foods may be chopped to a coarse, medium or fine size.
To mince is to reduce food 1/8 of an inch or less in size. Mincing doesn't require uniformity of shape, unlike dicing. Mincing may involve grating foods. As with the above two cutting methods, minces may be coarse, medium or fine.
Dicing, chopping and mincing foods release their flavors. Cutting foods into smaller pieces also increases their exposed surface area, allowing them to be cooked more evenly.
Chopping is used for ingredients that are intended to retain some of their character. This is why, for instance, you chop vegetables when you add them to soups or salads—you want the carrots or onions or tomatoes to flavor the overall dish, but you also want to still be able to experience them in individual chunks. Dicing is used when you still want to taste ingredients individually, but want them to have a milder impact than chopped ingredients. Finally, mincing is mostly used when preparing herbs or garnishes. Herbs are often very strongly flavored, so we want them to disappear into the overall dish, rather than remaining in concentrated chunks.