The term "bundle" is an abbreviated form of the full term fiber bundle. Depending on context, it may mean one of the special cases of fiber bundles, such as a vector bundle or a principal bundle. Bundles are so named because they contain a collection of objects which, like a bundle of hay, are held together in a special way. All of the fibers line up--or at least they line up to nearby fibers.
Locally, a bundle looks like a product manifold in a trivialization. The graph of
a function 
sits inside the product as 
.
The bundle sections of a bundle generalize functions
in this way. It is necessary to use bundles when the range of a function only makes
sense locally, as in the case of a vector field on
the sphere.
Bundles are a special kind of sheaf.
