in a professional kitchen, small dice is a specifcation for a knife cut. It means that you are to cut an item into perfect 1/4" cubes. Common items for small dice include onions, carrot, celery, turnip, potato, fennel, radish, and celeriac.
To execute a small dice, take your food item and form it into a rectangular block or tranche. Next, cut quarter inch planes working from one side to the other taking care to line up your knife properly at the top and bottom. A helpful tip to prevent your knife from sliding is to line the knife up and then start your cut with the tip of the blade. The angled cut allows the knife to move straight through the vegetable without gliding to the left or right during the cut. Once you have your 1/4" planes, you can now cut 1/4" batons long ways. For efficiency, stack the planes in groups of 2 or 3; whatever feels safe for you. Once all of your planes have been cut to batons. Work in groups to cut the batons into quarter inch cubes discarding any imperfect cuts and end pieces along the way.
As you work, have a container handy for any trim as well as a container for garbage (carrot peel for instance). It is always wise to assume that your chef will want you to keep any usable trim. The trim can be added to a stock or used in the preparation of family meal.
Other dice specifications include large dice - 3/4" by 3.4", medium dice - 1/2" by 1/2", and brunoise - 1/8" by 1/8". Mince and "chop" are not the same as dice.