Cutting the Deck is a way of reorganizing cards in the Deck. Cutting involves splitting the Deck into a few separate piles, most preferably two piles, then stacking the piles together differently to how they originally were. Since cutting does not change the relative order of the cards, it may not be used as a substitute for shuffling.
Before starting a Duel, the player must shuffle their own Deck, then their opponent must cut and shuffle it. During a Duel, if a player shuffles their Deck due to searching their Deck, their opponent has the opportunity to either shuffle or cut it.
In the anime and manga
In the anime and manga, during the Duel between Yami Yugi and Arkana, Arkana had used the stripper technique, which involved him trimming the edges of his "Dark Magician" cards, so when Yugi cut Arkana's Deck he would inadvertently cut it at the point so that the "Dark Magician" would be on top of a small stack, ensuring that one of the cards would be at the top of the entire Deck. Yugi, however, saw through this tactic due to Arkana using the card-damaging "shotgun shuffle" and devised a counter against it.