QUESTION: What is a PDF form?
ANSWER: A PDF form is a PDF file that a computer user can interact with. A PDF form contains boxes where a user can type text, lists from which a user can choose particular items, boxes a user can check, etc. IRS Form 941 is an example of a PDF form.
Creating PDF forms
There are various software applications that allow you to use a point-and-click interface to create PDF forms. Most people probably use a commercial version of the Adobe Acrobat software to create them. (When we say "a commercial version," we mean a version of Adobe Acrobat that costs a few hundred dollars. Note that the free Adobe Reader software cannot be used to create PDF forms.)
In addition to the applications mentioned above, there are various applications made by companies other than Adobe that allow you to create PDF forms.
Using PDF forms
The mechanisms that allow a user to interact with a PDF form are contained in some "PDF-viewer" applications. The Adobe Acrobat software and the Adobe Reader software are PDF-viewer applications, and most modern Web browsers have PDF-viewer capability.
When a users interacts with a PDF form, the data provided by the user can either be saved in the PDF form itself (the PDF form is after all a PDF file), or it can be stored on a server; it depends on how the workflow is set up.
In the former case, the user would click File > Save within the PDF-viewer application, would then specify a filename for the to-be-saved PDF file, and would then click Save.*
In the latter case, the PDF form would have a button that the user would click when he has finished filling out the form, and when the users clicks that button, the data provided by the user is transmitted to a server and saved there.
*Not all PDF-viewer applications allow the user to save a PDF form containing filled-in data.
Besides allowing a human user to interact with a PDF form, there are other ways to put information into the fields in a PDF form. An Internet search will lead you to programmer's toolkits/libraries that can facilitate development of a system that populates PDF forms dynamically.
See also What is an HTML form?
See also What is FDF?
See also What is XFDF?