This question comes out of the answers to <a href="https://mathoverflow.net/questions/7836/why-is-it-useful-to-study-vector-bundles">Ho Chung Siu's</a> question about vector bundles.  Based on my reading, it seems that the definition of the term "section" went through several phases of generality, starting with vector bundles and ending with **any** right inverse.  So admittedly I'm a little confused about which level of generality is the most useful.

Some specific questions:

- Why can we think of sections of a bundle on a space as generalized functions on the space?  (I'm being intentionally vague about the kind of bundle and the kind of space.)
- What's the relationship between sections of a bundle and sections of a sheaf?
- How should I think about right inverses in general?  I essentially only have intuition for the set-theoretic right inverse.

Pointers to resources instead of answers would also be great.