The problem is that your system doesn't know what program to use to open a .PAF file. Go to Start / Default Programs / Associate a File or Protocol with a Program and you'll see that either there is no .PAF there or there is no file associated with it. You need to change it so that there's an associated program if it exists or install a program to create the extension with the association.
You need to ask the person who sent it to you what program was used to create the file. It's probably a Personal Ancestral File but that's by no means certain. Once you know what program created the file, you can then have a chance to discover what program will open the file - certainly the same program that created it but perhaps some other programs as well. Adobe Acrobat costs money and makes .PDF files, but they can be read using Adobe Reader for free. Microsoft Word creates .DOCX files at a cost, but they can be read for free with Word Viewer or with OpenOffice. The same may apply here (or it may not).
In Spirit's google search, I saw at least three possible programs that could have created the file (and there were probably more) - the most likely being some Personal Ancestry program http://www.file-extensions.org/paf-file-extension. Some programs that can open a .PAF file are free and others cost money - it depends on what was used. Once you obtain and install the program, an association will be created and you'll be able to read .PAF files with the program (if that's associated with the program that created the .PAF in the first place).
Good luck and I hope this helps.
Lorien - One - MCSE/MCSA/Network+/A+