What are some easily-stated relatively recently proven theorems? I don't mean they were necessarily easy to prove, just easy to state. Here are a few examples:
The proof of Fermat's Last Theorem was completed in 1995, according to Wikipedia, by Wiles and others.
Green and Tao proved that for any $N$, there is an arithmetic sequence of primes of length at least $N$.
This year, Yitang Zhang made progress toward the twin prime conjecture - he found an integer $K$ such that there are infinitely many pairs of distinct primes that differ by less than $K$. I think that $K$ was in the neighborhood of 17 million or so but lower bounds were found within months. Sorry I don't have more specifics; see Yitang Zhang: Prime Gaps.
According to http://truthiscool.com/prime-numbers-the-271-year-old-puzzle-resolved, Helfgott has proved the weak Goldbach conjecture (any odd integer >5 is the sum of 3 primes (that is the wording from the article, I apologize if it is imprecise or wrong). The article states "Helfgott's preprint is endorsed and believed to be true by top mathematicians, Tao among them". The article is old (May 13, 2013) and I don't know if the result has been peer-reviewed and published in a journal. The conjecture is easy to state and if the proof is indeed valid it belongs on the list.
Notice that all four theorems above are in number theory (the statements of the theorems, anyway. The proofs may have used stuff from other branches of mathematics, I don't know.)
Fairly recently, some young folks found a deterministic primality-testing algorithm that had polynomial computational complexity (in time). One might consider this more of a theoretical computer science result than a mathematics result. Again, sorry, I forgot the specifics.
I think Perelman's proof of the Poincare Conjecture almost qualifies. It is difficult to explain exactly what it means for a manifold to be orientable, even to most mathematicians, let alone laymen.