The Liar Paradox is a logic problem that goes back (at least) as far as the Ancient Greeks.
Example :
A known liar, who never tells the truth, says "I am lying". If he is, as usual, lying, then his statement is false - meaning that he's telling the truth. Leading to a seemingly unsolvable paradoxical situation.
Another (famous) example of a similar paradox is "This statement is false"
Many modern-day philosophers have attempted to unravel the paradox, notably, Alfred Tarski, Arthur Prior, Saul Kripke, Jon Barwise, John Etchemendy, Graham Priest and Andrew Irvine.
Although some - e.g. Bertrand Russell - have pointed to the inherent 'vagueness' of language used to define and state the paradox :
Alfred Tarski showed that by using a vagueness-free formal language he could produce the Liar Paradox.
Source : The Internet Encylopedia of Philosphy
It's now generally accepted that there are ways out of the paradox - but philosophers can't agree on which is the appropriate solution.