First of all, the term "compact" is perhaps well-suited because any open cover has a finite ("small") sub-cover. But more generally, compactness generalizes certain properties of closed and bounded intervals $[a,b]$ in $\mathbb{R}$ to abstract topological spaces (and closed and bounded subsets of $\mathbb{R}^n$ are compact by the Bolzano–Weierstrass theorem; the Heine–Borel theorem applies to metric spaces), such as sequential compactness -- staying "trapped" within the set.

In particular, various notions of compactness can be helpful for dealing with e.g. function spaces. They again generalizes properties of the real line -- for instance, functions from $X$ (a compact metric space) to $Y$ (an arbitrary metric space) are uniformly continuous. The continuous image of a compact set is compact, just as in $\mathbb{R}$. Finally, we have the Stone-Weierstrass theorem for compact Hausdorff spaces and algebras of continuous functions, generalizing the Weierstrass approximation theorem for polynomials defined on closed intervals. There are many other theorems for abstract topological spaces where compactness is important.

Of course, these theorems apply to $\mathbb{R}^n$ as well!

impossible to do without them." (For Frechet, compactness meant what we'd call sequential compactness, which for your analysis reading is equivalent to the open cover version.) – KConrad Jun 30 '10 at 7:12