Definition: A collection $\lbrace G_\alpha\rbrace$ of open sets is said to be an open cover of the set $A$ if $A \subset \cup_\alpha G_\alpha$. A set $A$ is said to be compact if every open cover of $A$ contains a finite subcover.
In $\real^d$, we have the much simpler result that a set $A$ is compact if and only if $A$ is closed and bounded (in the sense that there exist $L, U: L \glt \x \glt U$ for all $\x \in A$). For example, the set $\lbrace\x: a_i \le x_i \le b_i$ for all $i\rbrace$, where $a_i < b_i$, is compact.
Definition: A function $g: \real^d \to \real$ has compact support if there exists a compact set $C$ such that $g(\x) = 0$ for all $\x \notin C$.
Consequences of compactness
Compactness is important because many important properties of continuous functions only hold when the domain is a compact set (i.e., the function has compact support):
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$g$ continuous $\implies g$ bounded
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$g$ continuous $\implies$ there exist $\a, \b: g(\a) = \inf g(\x)$, $g(\b) = \sup g(\x)$ (extreme value theorem)
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$g$ continuous $\implies g$ uniformly continuous