What exactly is RESTful programming?
I would say RESTful programming would be about creating systems (API) that follow the REST architectural style.
I found this fantastic, short, and easy to understand tutorial about REST by Dr. M. Elkstein and quoting the essential part that would answer your question for the most part:
Learn REST: A Tutorial
REST is an architecture style for designing networked applications. The idea is that, rather than using complex mechanisms such as CORBA, RPC or SOAP to connect between machines, simple HTTP is used to make calls between machines.
- In many ways, the World Wide Web itself, based on HTTP, can be viewed as a REST-based architecture.
RESTful applications use HTTP requests to post data (create and/or update), read data (e.g., make queries), and delete data. Thus, REST uses HTTP for all four CRUD (Create/Read/Update/Delete) operations.
I don't think you should feel stupid for not hearing about REST outside Stack Overflow..., I would be in the same situation!; answers to this other SO question on Why is REST getting big now could ease some feelings.