A computable number is a
number that can be computed. D'oh.
Of course, the much more interesting case is a number that can not be computed. This may seem illogical to the layman at first, because after all, computing numbers is what computers are all about. But what about non-finite, non-cyclic real numbers? Some of them, like pi and e, are computable. Of course they are infinitely long, so they cannot be computed fully, but we have well-defined methods to compute any finite prefix of them in a finite time.
What about the rest of them, those that don't have a spiffy algorithm to describe them?
Or better yet, what about integers we can specify but don't know the value of (and never will)? They exist.
For example, what about the number of steps a Turing machine takes to disprove Goldbach's conjecture? If it is untrue, it can be disproven by finding a counterexample through brute force. Unfortunately Godel's Theorem
ensures that there will always be statements which can neither be proven nor disproven, so that the number of steps the Turing machine takes will remain unknown forever. More generally, this is called the Halting Problem.