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 space of an open set in the base
and a fiber in a trivialization.
The function graph of a function 
sits inside the product
space 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.
The bundle sections over open sets in the base form a sheaf.
