There is a model of a computer that is used for this. It is called the Turing machine. A Turing machine basically is a special typewriter with an endless ribbon. The machine is named after the mathematician Alan Turing.
A problem is computable if it can be expressed in such a way that a Turing machine can solve it.
One of the best known examples is the Halting problem. The task is to write a program which says for all programs whether they will finally stop. This is impossible to decide. Mathematicians say the problem is undecidable.