I have been there, like so many of us. There are so many confusing words like Web API, REST, RESTful, HTTP, SOAP, WCF, Web Services... and many more around this topic. But I am going to give brief explanation of only those which you have asked.
It is neither an API nor a framework. It is just an architectural concept. You can find more details here.
I have not come across any formal definition of RESTful anywhere. I believe it is just another buzzword for APIs to say if they comply with REST specifications.
EDIT: There is another trending open source initiative OpenAPI Specification (OAS) (formerly known as Swagger) to standardise REST APIs.
It in an open source framework for writing HTTP APIs. These APIs can be RESTful or not. Most HTTP APIs we write are not RESTful. This framework implements HTTP protocol specification and hence you hear terms like URIs, request/response headers, caching, versioning, various content types(formats).
Note: I have not used the term Web Services deliberately because it is a confusing term to use. Some people use this as a generic concept, I preferred to call them HTTP APIs. There is an actual framework named 'Web Services' by Microsoft like Web API. However it implements another protocol called SOAP.