PDF forms are forms created in the Adobe Acrobat program. They are used as they will print the same on different printers and the information making the form is not easily changed.
For most users the answer is no you cannot save the information you enter in a pdf (Adobe Acrobat) form. Most people just have the Adobe Acrobat Reader program on their computer which is free and just allows them to view Adobe Acrobat forms. To be able to save the information you enter in an Adobe Acrobat pdf form you would either need the Adobe Acrobat Profession program or one of the third party programs that allows you to save that data.You can search the Internet to find these which include CutePDF and others). If you fill out a pdf type form you can either print it or if there is a send button at the top of it send the completed form (via email) to the creator of the form,.
Microsoft Office 2013 will allow you to edit and work with Adobe Acrobat forms and Microsoft Office 2010 allow you to create pdf forms (it is in the Save as options).
Adobe historically has kept the pdf file type very proprietary meaning they did not share out information about how it works internally to others.
Thanks Doris for the question