Classifying of fiber bundles is dependent on its structure group, which is a Lie group, essentially it is how those fibers are pasted together. For any Lie Group $G$, there is a classifying space $BG$ associated to this group. And the theorem says, bundles over a space $X$ (good enough) with given fiber $F$ (with good enough action) is one to one correspondence with the homotopy classes from $X$ to $BG$. This also leads to the definition of Characteristic classes, which is in some sense, just the pull back of the generator of the cohomology Ring for $BG$.