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Drop what you're doing usually happens during an opening shot. A cutscene begins and a character is patiently reading a nice book. Suddenly, his cousin comes in and tells him that something terrible has occurred. The character drops the book, not even folding the page so he can remember where he was, and runs off.
This type of situation is used to get the plot moving to interesting places. Usually, the main protagonist is involved in such a situation. But sometimes, this doesn't even happen in a cutscene, but to a regular bystander. For example, you, the player, fire your gun in the street. All the NPCs that were sweeping or minding their own business will instantly drop what they were doing and run for the hills. Although in some games no action is required to interrupt someone. This is probably most notable in the Fable series, where the crowd comes to the hero and starts cheering (If his alignment is good) or booing (If his alignment is evil). Most of the time, workers drop heavy crates that they were carrying and after the hero moves away from their sight, they proceed with their work (Going back to pick another crate etc.). This is becoming more and more popular, occurring in games like Grand Theft Auto IV, Mafia II and Assassin's Creed: Brotherhood, where people would drop things in shock or amusement and just stand there, watching the protagonist.
Not to be confused with occurrences where the player actually interacts with an NPC. Which would be the case in Assassin's Creed series, where the protagonist can even choose if he wants to perform a gentle push or knock them down.