No, they're not the same. You also forget the Core 2 Duo :)
1) processors are a little chip with some brains. Idea -- put two brains on one physical chip. Then, you get a 'Dual Core' processor. Intel makes them, AMD does too... it's just an idea, and not a product. Each brain is a core, so two of them on the same chip is called a dual core (helps to share the processing.. two brains are better than one...).
2) Intel Core Duo is one type of processor Intel makes that is dual core. It's only 32bit though. Intel Core 2 Duo offers slightly higher speeds, and is also 64bit. This may be useful if you need 64bit capabilities (you probably don't).
They do different sorts -- some for mobile uses, some more for desktop uses. You shouldn't really think too much about this though; the manufacturer will use which one it considers best.
3) Centrino is their moble line of processors, that use less power. Came out around the Pentium 4 time, and the Centrino Duo is the same but dual core.
Personally, I have a Core 2 Duo, and am happy with it. I'd rate it above the two you mention in your question.