Priggish people are snobby and self-righteous. An overly prim and proper movie character who's always telling other people what they should do is priggish.
If you offer your opinion on how your friends should live their lives, and especially if you're very uptight and snooty, people will think you're priggish. The adjective priggish comes from the eighteenth century prig, "precise in speech and manners," which was also used to mean "religiously devout." Earlier, a prig was "a petty thief."