Are there strong results in contemporary mathematical research (last 20 years) which have a proof which every mathematician (holding a PhD) can completely understand within a few days?  If yes, please give examples. If not, please explain the phenomenon.

How about the saturation conjecture? It is a quite strong result, and I believe the proof should be accessible within that time, see www.math.rutgers.edu/~asbuch/papers/sat.ps 


There are a number of such results in combinatorics. The Wilf–Zeilberger method is perhaps the "strongest" one that I can think of off the top of my head. It is strong by any reasonable definition of the word. With a more precise definition of "strong" I could probably come up with others. 


I think "Primes is in P" is accessible to everyone. 


My favorite from the last few years is the proof by Hugo DuminilCopin and Stanislav Smirnov that the connective constant of the honeycomb lattice is $\sqrt{2+\sqrt{2}}$: http://arxiv.org/abs/1007.0575 


Manjul Bhargava's proof of the 15 theorem (in Quadratic Forms and Their Applications, Contemp. Math. 272, 1999) is strikingly simple, especially when contrasted with Conway and Schneeberger's original, intricate proof. 

