What is the Church-Turing Thesis?

Simply stated, the Church-Turing Thesis says the following: Any algorithm can be implemented on a Turing Machine.

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There may be more than one way to solve a problem. Any one of those methods, as long as each step is logically correct, should give the same correct answer. The Church-Turing Thesis says that as long as you can somehow explain to someone how to get the answer, or as long as you can explain to a computer (by writing some computer program) on what to do to get the answer, then it is also possible to design a Turing Machine where in each of its states it will know what to do next to eventually get the same answer.

The Church-Turing Thesis is called a thesis, or conjecture, rather than a theorem, because neither Alonzo Church, nor his student Alan Turing, nor anyone else yet, have been able to prove it with a rigorous method worthy of a mathematical theorem--as was done, for example, with Gödel's Incompleteness Theorem.

However, the Church-Turing Thesis is generally accepted as true. No one has yet been able to come up with an algorithm that couldn't somehow be implemented on a Turing Machine.