One problem which I think is mentioned in Guy's book is the integer block problem: does there exist a cuboid (aka "brick") where the width, height, breadth, length of diagonals on each face, and the length of the main diagonal are all integers?
update 2012-07-12 Since the question has returned to the front page, I'm taking the liberty to add some links that I found after Scott Carnahan's comments. (Scott deserves the credit, really, but I thought the links belonged in the answer rather than in the comments.)
The Hot spot conjecture The conjecture seems quite amazing and simple to formulate,
it can be even understood by persons "from the street" seems its prediction can be tested experimentally. It is a subject of "polymath project 7".
Let me quote:
The hotspots conjecture can be expressed in simple English as:
Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.
In mathematical terms, we consider a two-dimensional bounded connected domain D and let u(x,t) (the heat at point x at time t) satisfy the heat equation with Neumann boundary conditions. We then conjecture that
For sufficiently large t > 0, u(x,t) achieves its maximum on the boundary of D
This conjecture has been proven for some domains and proven to be false for others. In particular it has been proven to be true for obtuse and right triangles, but the case of an acute triangle remains open. The proposal is that we prove the Hot Spots conjecture for acute triangles!
Note: strictly speaking, the conjecture is only believed to hold for generic solutions to the heat equation. As such, the conjecture is then equivalent to the assertion that the generic eigenvectors of the second eigenvalue of the Laplacian attain their maximum on the boundary.
A stronger version of the conjecture asserts that
For all non-equilateral acute triangles, the second Neumann eigenvalue is simple;
and
The second Neumann eigenfunction attains its extrema only at the boundary of the triangle.
(In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.)
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May be this problem can be mentioned when teaching determinants and in particular:
$\det(AB)= \det(A)\det(B).$
There are so-called Capelli identities which generalize this formula for specific matrices with non-commutative entries.
In the paper Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities by Sergio Caracciolo, Andrea Sportiello, Alan D. Sokal they formulate
certain conjectures of the type $$\det(A)\det(B)=\det(AB+\text{correction})$$
on the page 36 (bottom), conjectures 5.1, 5.2.
I think these are quite non-trivial, but probably some smart young mathematician may solve them,
given some amount of time (some months may be).
I spent some amount of time thinking on them without success, and moreover
let me mention that D. Zeileberger and D. Foata also
failed to find a combinatorial proof of the Capelli identity of very similar type --
the one proved by Kostant-Sahi and Howe-Umeda --
see their comments in Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory page 9 bottom: "Although we are unable to prove the above identity combinatorially ... ".
So words above are some idications of non-triviality of the conjectures.
Personally I am quite interested in a proof, probably it can give clue for further generalizations.
Best Answer
I'd nominate the theory of Macdonald polynomials (and associated topics). This is an extremely important area of algebraic combinatorics. Even if we restrict to type A, there are certain features of the subject that are intrinsically complicated, but nevertheless, learning this subject is currently a lot harder than it needs to be IMO. Haglund's monograph is certainly helpful because it gives a clear account of a lot of the purely combinatorial side of the theory, but his goal was not to give a full account of Macdonald polynomials.
I recently discovered the slides for a talk by Ole Warnaar that give a very nice introduction. They solve an important aspect of the exposition problem, which is to construct an engaging story that can serve as the backbone for a more complete exposition. Of course the slides themselves only scratch the surface.