Core 2 Duo are built on a more advanced processor technology (45nm thick). Core Duo are built older tech that is 65nm thick.
The result is that for a given clock speed (e.g. 2.0), Core 2 will run faster and cooler than a Core Duo
Duo means 2 processors per chip, Quad means 4 processors per chip, so will run programs that can spread their processes across the 4 processors a lot faster.
Core 2 Quad is ideal for intensive video processing, photo editing or gaming, but is a bit of overkill for routine tasks such as Word, internet etc.
Core 2 Duo is ideal for laptops as it uses a bit less power, and will give a slightly better battery life, while not being as warm on your lap.
Core Duo is ideal for a budget computer as it's cheaper.