A flag is a persistent indicator to the sniper that a mission has occurred. A flag is similar to a tell, except that a tell can only be seen at the time it happens, whereas a flag continues to be visible afterward. Currently, there are only two missions which have a flag.
If the guest list is missing from Toby's tray/pocket, this indicates that the Purloin Guest List mission has been performed. The flag can be reset by returning the guest list, potentially preventing the sniper from noticing that the purloin has occurred.
The flag for the Swap Statue mission is a statue that has changed to another model. This depends on the sniper memorizing the statue models before the spy performs the swap. The flag cannot be reset.