Some PDF forms can be saved, including all filled-in field data:
Some others can not be saved, and all filled-in field data are lost:
In Acrobat, click on the "Advanced" tab, then click on "Enable Features in Adobe Reader." This should do the trick.
There is a setting inside the PDF file that turns on the allow saving with data bit. However, it requires that you have a copy of Adobe Acrobat installed to change the bit.
The only other option is to print it to a PDF print driver which would save the data merged with the pdf file.
UPDATE: The relevant information from adobe is at: http://www.adobeforums.com/webx?13@@.3bbb313f/7
In Acrobat XI, (Close Form Editing if open) File > Save As Other > Reader Extended PDF > Enable Additional Features
If you are using Adobe Acrobat X to make the form, set all the fields as you want them, then click File, Save As, Reader Extended PDF, Enable Additional Features. The resulting PDF form can be saved when filled in, if opened in versions of Adobe Reader before XI.