The most familiar example is 2-form electromagnetism. This is a modified version of Maxwell's equations in which the connection 1-form is replaced by a 2-form, sometimes called the Kalb-Ramond field. Just as you can integrate the A field over the worldline of a particle to get a term in the action which describes its coupling to the electromagnetic field, you can integrate the B field over the worldsheet of a string to get a term in the action which describes its coupling to the Kalb-Ramond field! But the really interesting thing is that you can think of the Kalb-Ramond field as a connection on a "principal 2-bundle" where the relevant gauge "2-group" has only one object, but a U(1)'s worth of morphisms.
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 from 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. I should also create lower-resolution files that are quicker to download. Of course, if anyone else wants to do this, I'll be eternally grateful, and my thanks will appear here.
If you want to dig deeper into this subject, try the Spring 2002 notes!