Ok, so title sounds horrible cause it wouldn't fit.
What specs do i need to know about to see if the ram would be swappable? Will a quad core work with 6 GB of ram down from 8gb? Will a dual core work with 3 gb ram? Dual core is pentium D, quad is q8200
Check your Motherboard Specs and Manual.
Make sure the RAM is DDR2
It might WORK but be sure both Motherboards support the same DDR's. ex it will not work if your ram from the Pentium is DDR but the Quad core Motherboard only supports DDR2.