My top dozen candidates that are relatively easy for non-mathematicians to understand (this would probably change if I thought about it more... and this is in no particular order):

1. The Central Limit Theorem (which proves that under some very general conditions, a lot of random distributions add up to a normal distribution)
2. The Fundamental Theorem of Algebra (which tells you that an algebraic equation of order n has n roots)
3. The Jordan Curve Theorem (which proves that a closed curve on the plane has an inside and an outside, something you probably didn't think needed proving)
4. The Pythagoras Theorem (it may be simple and high-school level, but the entire field of trigonometry and analytical geometry relies on it).
5. The Nash embedding theorem, (which I won't attempt to explain in a parenthetical remark: http://en.wikipedia.org/wiki/Nas... )
6. The proof that the number of primes is infinite (this is old enough that I don't think it is attributable to any one person)
7. The proof of Euler's formula (I feel like applauding this particular beast)
8. Godel's Incompleteness Theorems (which show that that any logical system at least as complex as arithmetic contains true statements that are unprovable within the system)
9. Brouwer fixed-point theorem (another one which cannot be summarized in a parenthetical remark: http://en.wikipedia.org/wiki/Bro... ... but basically most of differential equation theory rests on this)
10. No-free-lunch theorems in combinatorial optimization (crudely, any optimization procedure that works better than another in one part of a problem space will perform worse in another part, but that is an oversimplification: http://en.wikipedia.org/wiki/No_... )
11. The Binomial Theorem (Newton)
12. Von Neumann's demonstration of the universal constructor (not really a theorem per se, but a proof by construction that self-replication was possible in cellular automata, which to me is a sort of "proof that life itself is a mathematical thing." http://en.wikipedia.org/wiki/Von... )

There are a lot more theorems I really like, but many of them (including extremely important ones that affect your life) cannot even be stated without introducing several layers of non-trivial concepts first.

These dozen aren't necessarily the most important theorems in mathematics, but they are the ones whose importance is the easiest to understand (with the exception of the Nash embedding theorem, which you probably won't understand if you haven't had some topology, but I like it so much I decided to put it in anyway).

For the record, I only understand the proofs of about half of my list.