I want to upgrade the mother board in my Vaio PCV-RZ44G Desktop so that I can run Windows 8. What board forms are compatible and is an Intel I3 or I5 CPU better?

You would be better off applying the money to a new computer. You are opening "Pandora's Box" . You will end up with a old case with ALL new components. That may or may not work, A.K.A., "pain and heartache".

I'm currrently owning that sony vaio pcv-rz44g but I could not upgrade my agp graphics card in year 2007 since sony did not allow us to upgrade their 300w PSU. The sony psu size was not standard so it is not upgradable. However, very recently I was experimenting on building a new very cheap pc rig since the newer cheap Lenovo all-in-one pc had malfunctioned while its core i3 processor was still working so I decided to save money and put an entire new entry level Intel Haswell motherboard with 2 PCI sockets (for Sony video in/out card and older pci sound card) and an old VGA connector (for old sony monitor) into this sony vaio case. The Gigabyte GA-H81M-S2PV or GA-H81M-S2PT or GA-H81M-HD3 motherboard would be perfect match. The sony vaio's old motherboard is a Micro ATX form factor motherboard so only a new micro atx motherboard or perhaps even a mini itx motherboard would fit into this old vaio case and it must also be an entry level core i3 system since the PSU is only at max 300w and it could not be upgrade. If it still does not fit then remove the rear metallic panel or even worse we might have to cut the rear metallic plate. This could be a re-modified old pc case for very tight budget people. Keep in mind that the upgradability of this very old pc is very limited and Sony would not even allow us to upgrade their PSU so this is why they chose a non standard size PSU. Don't even bother thinking about adding a discreet performance level graphics card or overclocking the cpu and the ram. Please note that this is good only for those who are very tight on a budget and does not care much about future upgrade but in the end, I would still recommend you to go and buy a new pc case which should not cost a lot and being compatible for both standard atx and micro atx motherboards or just build the entire new pc rig.