Usually it's a type of web service or API, which you can interface with to get data from external systems into your system.
RESTful is opposed to something like SOAP. They're both usually web services, but they work differently. The big key with RESTful web services is they are "stateless", which means each transaction between it is a complete entity all on it's own. (As opposed to SOAP where you may do something like login, then have a persisting connection).
The other big thing is it usually uses the HTTP layer as it's meant to be used. A lot of APIs have you do something like get your info from a URL:
A RESTful API would have you put this as a URL usually
Which translates into a HTTP GET request:
HTTP 1.1 GET /api/books/fun/10
That's not much different. But say you want to create a new user. A non-RESTful API may have you do something like:
which translates to:
HTTP 1.1 GET /makeuser.php?name=bob&password=tooth
However, that's not using HTTP GET properly, since GET should only get info and never change it.
A proper RESTful API would likely have you use PUT instead. Something like:
HTTP 1.1 PUT /makeuser/bob/tooth
Making proper use of the HTTP protocol.