The primary difference rests with the what the final dish will look like e.g. Presentation. Cubing/dicing result in a visible element in the presentation, while chopping and mincing are intended to almost make the item disappear and expose as much surface as possible to extract the flavors into the remainder of the preparation.
DICING: The largest of the group refers to uniformly sized pieces, usually cubes. ¼" is pretty much standard size. A large dice would be anything bigger than that. However anything in the 3/4" to 1" is just "Cubing"
BRUNOISE: Is also a cubing style, but in the ⅛" size typical of French cooking.
MINCING: Again we are looking for a very finely irregular cut that results in almost a paste or course powder. Again uniformity shape is not important, just the fineness.
Tips:
This may not be the appropriate place for this, but when chopping or mincing garlic or ginger, peel it first, then give it a good whack with the side of your chef's knife or cleaver. It will quickly reduce either to a more manageable shape. However if you are too vehement with the "whack" they do tend to spatter.