This question comes out of the answers to Ho Chung Siu's 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.