When you trounce someone, you win decisively. If you win a chess match in three moves, you trounce your opponent.
When a professional tennis player is in the midst of a tournament, she hopes to trounce her rival, and an ambitious speller might dream of the day she competes nationally and trounces all the other competitors. A victorious army can also be said to trounce the enemy. The origin of trounce is uncertain, though one theory has the French troncer, or "cut a piece off from," as its root.