What is JSON?
By defenition JSON is:

"JSON (an acronym for JavaScript Object Notation) is a lightweight text-based open standard designed for human-readable data interchange. It is derived from the JavaScript programming language for representing simple data structures and associative arrays, called objects."

It is a great way to send/tansmit data over a network connection. With it we can send a request to a server application and get a response back in real time.
Great, so this JSONP thing then...
JSON only allows connections and requests to be made if the server is on the same domain. JSONP allows us to make data requestions to servers on different domains, aka cross domain access. Both JSON and JSONP are useful because we can write this communication technique in Javascript and use it directly in a web browser.
The third wheel, Ajax.
Ajax is similar to a JSONP call, but it allows for a more dynamic user experience. We can use AJAX for any asynchronous request made by the browser, or anything that uses "XmlHttpRequest".