Will ubuntu work on the Acer Aspire One 756 with the Pentium B967 processor, and if so, what version should be used?
Yes, Ubuntu should work for your computer. It's hard to say for sure until you have tried it, but I can't think of any reason why it would not work. I would say to try 12.04 if you can. I am assuming that this computer is a netbook from the information I have seen about it online. As such, you may have to use the Ubuntu alternate text-based installer. You can find this on their website. http://www.ubuntu.com/download/desktop/alternative-downloads
For more specifics on whether or not this will work on your laptop, here's a link to a forum talking about it: http://ubuntuforums.org/showthread.php?t=2041487
I hope this helped you. I hope Ubuntu works on your laptop.