When we say "My money is on something," we mean we are willing to bet that that something will win or be the actual outcome. "My money is on the Blackhawks" means I think the Blackhawks will win the Stanley Cup.
"Tuck and roll" basically means to do a somersault.
The quote you are asking about is from the TV show "Big Bang Theory." You can find the quote, in context, on this page:
It is in Quote #6. Leonard responds to the two possible outcomes that Howard mentions by saying "My money's on 'tuck and roll.'" This is the outcome that Leonard thinks will happen.