An example of an easy to understand undecidable problem

"Are these two real numbers (or functions, or grammars, or mathematical statements) equivalent?"
(See also word problem)

"Does this statement follow from these axioms?"
(Hilbert's Entscheidungsproblem)

"Does this computer program ever stop?"
"Does this computer program have any security vulnerabilities?"
"Does this computer program do <any non-trivial statement>?"
(The halting-problem, from which all semantic properties can be reduced)

"Can this set of domino-like tiles tile the plane?"
(See Tiling Problem)

"Does this Diophantine equation have an integer solution?"
(See Hilbert's Tenth Problem)

"Given two lists of strings, is there a list of indices such that the concatenations from both lists are equal?"
(See Post correspondence problem)

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There is also a large list on wikipedia.

I think the Post correspondence problem is a very good example of a simple undecideable problem that is also relatively unknown.

Given a finite set of string tuples

(a , bba) X
(ab,  aa) Y
(bba, bb) Z

the problem is to determine if there is a finite sequence of these tuples , allowing for repetition, such that the concatenation of the first half is equal to the concatenation of second half

(bba, bb) Z
(ab,  aa) Y
(bba, bb) Z
(a,  bba) X
------------ gives
(bbaabbbaa, bbaabbbaa)

The only big issue I have with this problem is that the only undecideability proof I know of falls back on simulating a Turing machine - it would be nice to find a more elementary alternate version.

May be you want to check these:

Alan_Turing_and_Undecidable_Problems_in_Mathematics on fora.tv

what-are-the-most-attractive-turing-undecidable-problems-in-mathematics on mathoverflow

MagicSquare on mathworld.wolfram