To mince an onion, first cut it in half from root to tip and peel it. Lay one half on its flat side--this way it won't roll around the board.
Slice down vertically, from the root end down, making as many parallel slices as you can. Do not cut through the root, though, since that is what holds the onion layers together.
Then, holding the blade horizontally, cut through the onion several more times.
This makes a grid within the onion that you can cut across to create very small pieces. The same technique can be used on garlic, shallots, tomatoes or any hard vegetable.