I thought I'd give the experts on this subject a day to respond. But as they haven't, I'll describe this nice computational application of compactness myself:
One interesting feature of compact spaces is that they can be exhaustively searched on a computer in a finite time, even if they are infinite.
A great example is described here. Consider the Cantor space of infinite binary sequences $C=2^\omega$. Suppose we have a computable predicate on this space, ie. a computable function $f\colon 2^\omega\rightarrow 2=\{0,1\}$. Then we can search all of $C$ to find an element that satisfies $f(x)=1$, or show that there is no such $x$, with an algorithm that is guaranteed to terminate.
The algorithm is described here. The important point is that (with the right topology) the computable functions are continuous and that the Cantor space is compact. This means that any predicate $f$ on $C$ is uniformly continuous in the sense that there is an $n$ such that $f$ can be computed without examining more than the first $n$ digits in its argument. From that it can eventually be deduced that the search for an element of $C$ satisfying $f$ can be completed in finite time.
This is somewhat surprising given that we can't exhaustively search $\mathbb{N}$ with a computable predicate in finite time, and yet $2^\omega$ seems like a 'larger' space.
