The words "dual core" is a general term, which means that you have a processor with two processing cores.
Intel Core Duo is a specific processor architecture from Intel. Intel Core 2 Duo is another specific processor architecture from Intel, newer than Core Duo.
There are differences between the capabilities of Intel Core Duo and Core 2 Duo processors. One of them is that Core Duo processors are 32-bit only, while Core 2 Duo processors support 32-bit as well as 64-bit operating systems.
The Core i3, i5, i7 series of processors is what comes after the Core 2 Duo series. These processors have a newer architecture and are again faster than Core 2 Duo processors. There's also a second generation of Core i3, i5, and i7 processors (the ones that have product codes with four digits, starting with a 2, for example Core i5-2500, Core i7-2600, etc.).
It's a big jungle of product names and codes. If you want to understand all of it, the Wikipedia article is a good starting point.