If someone cancels your card play with a "Stop Time!", your turn ends immediately.
The card you were going to use goes into the discard pile, and you don’t get to do anything else.
If you play "On the Move" and someone cancels it with a Stop Time, and you were also using a Move or Jump card to control where the Time Breaker was going, you don’t have to discard that card, only the On the Move card itself.
If someone plays "Quantum Leap", you can stop the entire action if you play a Stop Time card immediately.
However, if you delay until they've revealed another card, you must wait until they've revealed both of the cards they'll be playing and chosen which they will play first. You can then cancel either card, but not both (unless you have two Stop Time cards).