A dialetheia is a true contradiction - a statement such that both the statement and its negation are true. People who can stomach this idea, and believe there are such things, are called dialetheists - which doesn't imply anything about their religion: the word breaks down as di- (two) - aletheia (old Greek for truth - see aletheia where it is explained rather well).

A commonly given example is the 'liar paradox' (Epimenides paradox, 'this sentence is false', etc.) The reasoning may go as follows: if it's true, it's false; if false then true; hence assuming either its truth or falsity implies both its truth and its falsity - it comes out the same.

Disregarding the falsities, both the statement and its contradiction may be said (by brave dialetheists, anyway) to be true.

For this idea to be any use in logic, we must use a deductive system where a true contradiction does not let you immediately prove all statements (ie. the Ex Falso Quodlibet rule, which is valid in the 'normal' propositional calculus, does not hold) otherwise the acceptance of one dialetheia would enable us to prove any arbitrary statement, making our 'logic' trivial - any statement can be proved in it!

Fortunately help is to hand, in the form of paraconsistent logics and the related (and gloriously named) field of inconsistent mathematics1.

Paraconsistent logics which accomodate this logical 'feature' include many-valued logic(s) where statements can evaluate to true, false, or both true and false (to {T}, {F} or {T,F}) and worse.

The most plausible type of dialetheia I've found is not drawn from maths or logic. It's a quote from an obscure book called The third wor*d war. The original goes (as well as I can remember):

Words mean anything these days

Words do not mean anything these days

If that seems too self-referential or general to be plausible, how about:
Moral terms mean anything these days

Moral terms do not mean anything these days

Perhaps these are a special brand of dialetheia, such that the statement means the same thing as its contradiction. I doubt even paraconsistent logics have a theory for that, yet.

1. I swear I am not making this up. If you don't believe me, have a look at:
http://plato.stanford.edu/entries/mathematics-inconsistent/.