I understand JSON, but not JSONP. Wikipedia's document on JSON is (was) the top search result for JSONP. It says this:
JSONP or "JSON with padding" is a JSON extension wherein a prefix is specified as an input argument of the call itself.
Huh? What call? That doesn't make any sense to me. JSON is a data format. There's no call.
The 2nd search result is from some guy named Remy, who writes this about JSONP:
JSONP is script tag injection, passing the response from the server in to a user specified function.
I can sort of understand that, but it's still not making any sense.
So what is JSONP? Why was it created (what problem does it solve)? And why would I use it?
Addendum: I've just created a new page for JSONP on Wikipedia; it now has a clear and thorough description of JSONP, based on jvenema's answer.