Going back to Fermats theorem, the problem itself it really easy to understand, its the solution which is the problem
For reference and because its not been included abouve in this thread the problem is this:
Prove that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2
Because of the >2 element there are infinity options to factor which under current thinking at leat is impossible and so proving definitively that the statement is true requires a workaround of mathematics.
The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England the first on Monday 21 June, the second on Tuesday 22 June. In the final lecture on Wednesday 23 June 1993 at around 10.30 in the morning Wiles announced his proof of Fermat's Last Theorem as a corollary to his main results. Having written the theorem on the blackboard he said I will stop here and sat down. In fact Wiles had proved the Shimura-Taniyama-Weil Conjecture for a class of examples, including those necessary to prove Fermat's Last Theorem.
This, however, is not the end of the story. On 4 December 1993 Andrew Wiles made a statement in view of the speculation. He said that during the reviewing process a number of problems had emerged, most of which had been resolved. However one problem remains and Wiles essentially withdrew his claim to have a proof.
In fact, from the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties.
Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck.
In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.
On 6 October Wiles sent the new proof to three colleagues including Faltings. All liked the new proof which was essentially simpler than the earlier one. Faltings sent a simplification of part of the proof.
No proof of the complexity of this can easily be guaranteed to be correct, so a very small doubt will remain for some time. However when Taylor lectured at the British Mathematical Colloquium in Edinburgh in April 1995 he gave the impression that no real doubts remained over Fermat's Last Theorem.
Simon Singh wrote a book cunningly titled Fermats last theorem which is a great read for non mathematicians like myself, and having read it recently I thoroughly recommend it. It'll take you on a journey of discovery through mathematical history.