The Heine-Borel theorem is a fundamental result in real analysis that characterizes compact subsets of Euclidean space. It states that a subset of R^n is compact if and only if it is both closed and bounded. This means that the set contains all its limit points and fits within some finite region of space. Compactness is an important property in mathematics because it allows for the generalization of various theorems, such as the Extreme Value Theorem. The Heine-Borel theorem helps in understanding the behavior of functions and sequences within compact sets, making it a key concept in topology and analysis.