If you want to save the PDF forms along with the filled data you can:
Use any form with the paid Acrobat Professional
Use a form extended with Acrobat Professional (limited to 500 forms). See this article for more details.
Use the forms extended with LiveCycle Rights Extensions (unlimited distribution of forms is allowed). See this page for more details.
Using LiveCycle Reader Extensions a form designer can extend or apply document rights to the form to enable saving of filled data using even the free Reader. Using LC RE a PDF producer can achieve much more functionality for free Adobe Reader like, enabling end users to save data locally, add and save annotations, apply digital signatures, dynamically embed data in a barcode, and integrate with web services for PDF forms and documents.