The in-depth version seems to be well covered, maybe you're looking for the short-and-simplified version?
JSON is basically just a way to pass an array from one language to another.
It's used a lot for Ajax (amongst other things) because with Ajax you typically have a server-side language (PHP etc.) passing a set of results to a client-side language (javascript). Your javascript calls your PHP page with some parameters; your PHP page builds an array and echos it encodes it to JSON format; your javascript catches the JSON and decodes it back to an array to process.
There's more to it than that obviously (and for that reason I'm expecting a flurry of tear-streaked downvotes :) ), but that's all you need to get up and running with it.
