A fibration with the homotopy lifting property for CW complexes is often called fibration in the sense of Serre, in honour of the part played by the concept in the thesis of Jean-Pierre Serre. This established in algebraic topology the notion of spectral sequence, in a way that separated it out from the notion of sheaf (both concepts together having been implicit in the pioneer treatment of Jean Leray). Because a sheaf is 'as good as' a local homeomorphism, the notions seemed closely interlinked at the time.
The geometry of a fibration is, by definition, to do with being able to lift up homotopies in the base X into the fibering space Y. The fibers are by definition the subspaces of Y that are the inverse images of points x of X. We are not in this case given a local cartesian product structure (which defines the more restricted fiber bundle case), but something possibly weaker that still allows 'sideways' movement from fiber to fiber. One of the main desirable properties of the Serre spectral sequence is to account for the action of the fundamental group of the base X on the homology of the total space Y.
For the category theory notion of fibration, which is much more general, see semidirect product.
This article is a stub. You can help Wikipedia by fixing it.