a combination to the the words "fuck" and "shiznit," used as an exclamation to describe something which can be both good and bad for the person making the exclamation.
Aw fiznit, I gotta drive all the way to the bank to cash my paycheck.
The Rule is an acknowledgment of one's own mortality and is invoked as a means of avoiding activities, people, movies, organizations, books and TV shows that are very likely a total waste of time.
In essence, it means that each of us only have a finite number of summers left, and those summers aren't to be frittered away on meaningless crap.
"No, I didn't see the Hannah Montana movie; it runs counter to the Rule of Finite Summers."