- "Core 2" with a capital "C" is a model line of Intel processor, just like Pentium II was a model line of Intel processor. The Core 2 came after the Core notebook processor, just like the Pentium II came after the Pentium, so that's where the "2" came from. It's just a brand name.
- A dual-core processor simply has two processor cores. A core with a little "c" is a computing term referring to a part of a CPU that independently executes instructions. Intel and other vendors such as AMD sell dual-core processors.
- "Duo" was Intel's way of saying that a Core or Core 2-branded CPU had two cores. "Solo" meant one core, "Duo" meant two cores, and "Quad" meant four cores.
So if you understand what I wrote, you should be able to figure out that a Core 2 Duo is a particular model line of Intel dual-core CPUs.
Intel had lots of ways of saying how many CPU cores their CPUs had. They used the Solo, Duo, and Quad names for their Core and Core 2 CPUs, while they called dual-core Pentium 4 processors a "Pentium D," where "D" stood for dual-core. However, the "D" in the Celeron D CPUs selling at the same time as the Pentium Ds stood for "desktop" and they were only single-core CPUs. Intel also simply calls some processors "Dual Core," such as the Pentium Dual Core and Celeron Dual Core CPUs. Intel has since moved away from actually putting the number of cores in the brand name of the CPU- you get a brand name and a model number (such as "Core i7 980X") and then are left to look up the specifications of the CPU to see how many cores it has.
On the other hand, AMD simply puts an "X" and the number of cores in the CPU in the model name of desktop CPUs. Every AMD desktop CPU with an "X2" in the name is a dual-core CPU and every "X4" is a quad-core CPU. They have adhered pretty strongly to this nomenclature throughout their entire line of multi-core CPUs.