"Dual core" generally referes to dual core processors in general, that is, processors with two physical cores.
"Core Duo" is...well I think you mean "Core 2 Duo" which is the name (mostly marketing BS, nothing special) Intel has given to their line of Conroe and Wolfdale class dual core CPU's.
You obviously (no offense), don't really know what your talking about, I think it could be more productive to simply post the whole context of the question.
They all are just updates of each other better speed and power efficiency. Core 2 Duo is the latest get yourself a Core 2 Duo, they are great for all uses, such as gaming and running lots of applications.