Coarse chopping is generally used for mirepoix or similar flavoring ingredients that are to be strained out of the dish and discarded. It is also appropriate when cutting vegetables that will be puréed. Trim the root and stem ends and peel the vegetables if necessary. Slice or cut through the vegetables at nearly regular intervals until the cuts are relatively uniform. This need not be a perfectly neat cut, but all the pieces should be roughly the same size.
Mincing is a very fine cut, suitable for many vegetables and herbs. Onions, garlic, and shallots are often minced.