If you're the sort of mathematician who is fond of "stacks", you'll be pleased to know that just as an ordinary bundle has a sheaf of sections, a 2-bundle has a stack of sections.
If you're more interested in physics, you should think of 2-bundles as permitting a generalization of gauge theory that describes not the parallel transport of point particles but of 1-dimensional extended objects - open strings, or spin network edges. The most familiar example is 2-form electromagnetism, which I explained in detail in the Winter 2002 notes.
To get a general overview of these ideas, you should start by peeking at the slides of this talk:
The course notes below were taken by Alissa Crans, who has kindly scanned them in and made PDF files out of them:
Eventually I will try to give descriptive titles for the notes from each class, but right now I only have them listed by date.