(Sorry, last soft question!) Borwien, in The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, (probably quite rightly) says of the Riemann Hypothesis, that
No layman has ever been able to understand it and no mathematician has ever proved it.
Aside from being one of the most famous problems in mathematics, is this the most "difficult" to understand (by either layman of mathematician!) of all mathematical problems, or are others more "complicated" conceptually?
I realise that this is a highly subjective question (and perhaps one that is not worthy of this site), but I am interested in other problems that are perceived as being "difficult". I realise, of course, that the Millenium problems (themselves, based on the Hilbert problems, I believe) are all strong contenders. But are these the most " difficult" to understand?
For me, the axiom of choice, any problem associated with Cantor, or any definition involved in higher dimensions, such as Hilbert spaces, quaternions, etc. present considerable conceptual barriers, to mention just a few!
Here is the list of Millenium prize problems listed in order the difficulty in understanding their statements. Here a "understand X" does not include "appreciate significance" or "be able to read research papers papers discussing X". Some of the ordering is a bit random (problems 1 through 3 can be listed in any order.)
RH is far from the "most difficult to understand" mathematical problems. If you took, say, undergraduate complex analysis, you can understand the statement of RH simply by reading the Wikipedia article. This, in part, explain why it attracts so many mathematical cranks. Will's comments will tell you how to get a reformulation of RH without even undergraduate complex analysis, undergraduate calculus would suffice here.
Similarly, in order to understand the problem about NS equation, all you need is an undergraduate class in partial differential equations (and reading Wikipedia article). (Strangely, cranks seem to ignore this one, I guess they just do not like PDEs.)
Same goes for P vs NP, you just need good undergraduate-level CS courses to get this one.
To understand 3-d Poincare conjecture, you need to take a good upper division topology class (which gets you to the definitions of manifolds and the fundamental groups; many top-tier universities teach such classes). If not, then one semester of a standard graduate algebraic topology class will suffice here.
YM-existence and mass gap problem would require you to take a graduate course in differential geometry or topology (bundles, connections, etc).
Hodge conjecture can be understood after taking graduate courses in complex analysis, differential topology and algebraic topology (no algebraic geometry is really needed here even though the problem comes from algebraic geometry). OK, I am assuming that your courses in differential topology and complex analysis cover Hodge theorem and definition of (anti)holomorphic forms in several complex variables. It is a bit of a stretch, but you get it in better places.
Birch and Swinnerton-Dyer (BSD) Conjecture would be the hardest. Only a handful of math departments would offer graduate number theory courses covering L-functions.
However, anybody who got through the first few years of graduate school (in pure math) and is willing to spend few hours (say, less than 10) reading some unfamiliar material, can understand (the statements of) all the Millenium Problems. Does it mean that, say, BSD problem is the "most difficult to understand in the entire mathematics"? Not at all. There are many (less famous) problems in a variety of area of mathematics whose understanding would require much greater time investment (on the scale of years, not hours, unless you happen to specialize in that particular area, of course). For instance, just take a look at this list of open problems in algebraic topology. Or here for problems in model theory.