in a way, you are right. each core is a cpu, and the dual or quad just means how many cores are on the main processor. Intel started using the term core duo, then it went to core2duo which meant about the same. the core2quud is the 4 core model. All of these used the LGA 775 socket, and those motherboards used DDR2 RAM. AMD is not as clear but follows the same patterns. The newer i series went to DDR3 RAM, and there are 2 sockets used. LGA 1156 use dual channel DDR3 and the LGA 1366 use triple channel DDR3.
So if you want to stay with the DDR2 RAM, You can use a LGA 775 Motherboard ( I think that is what you have now.) 4 sticks of RAM should work, and will be fine for dual boxing.