RESTful is the native API of web browsers. When you put some website’s address into a browser, that’s an implied REST expression called a “GET” of the resource at that address. In response to that GET request, the web server on the other end returns a web page. However, REST is much more than requesting the resource (data) at some address. Just like using any website, one is able to Create things, Retrieve them afterwards, perform Updates to them, and eventually Delete them. That Create -> Retrieve -> Update -> Delete cycle is called “doing CRUD” (really), and that in a nutshell is what creating and using a RESTful system is all about.
In the “early days” of the Internet, when someone wanted to make a printer or some other machine programmatically communicate over the Internet, more complex systems with names like SOAP, XMLRPC and AMF were used to handle that communication. Then around the year 2000, a smart guy named Roy Fielding pointed out that the web itself was an API and these complex systems were not only a bother to create and work through, but needless because what they were offering was already built into the web itself.
Now, Drupal is a content management framework whose essential purpose is to create a website of some sort. You are probably familiar with some websites including information from other websites, such as a Twitter feed or Facebook friend status. This including of other website’s information can be accomplished “the old, hard way” via scraping the page that normally shows this data, via SOAP/XMLRPC or that communicating of information can be accomplished “the new, shiny RESTful way” which takes less effort and by it’s nature is universally supported.
This is essentially machine-to-machine communications, and is how an iPhone/iPad/Android/game console/printer or virtually any other device communicates on the Internet. This is using REST.