1. Name of chain of museums, books, and TV show by amateur anthropologist Robert Ripley about things which are odd, unusual, or surprising.
2. In colloquial conversation, an event which is surprising or which seems unlikely but is (supposedly) true.
Def 1 examples:
a man with father, grandfather, and great-grandfather all with the same birthday, or
Hindu priest who can drive a six inch metal spike into his nose all the way into his brain.
Def 2 examples:
Believe it or not, we are having twins! AGAIN!!
Believe it or not, she was choking, I was just performing the Heimlich maneuver!!
Believe it or not, I got the highest mark in the class!