Intel decided that they needed a new name after "Pentium" - they chose "Core" as something that would sound cool. When they adopted AMD's 64-bit instruction set, they decided to update the name to reflect a new generation of processors, and they changed it to "Core 2".
That's all that "Core" (with a capital "C") and "Core 2" are - marketing names for lines of processors. Both "Core" and "Core 2" processor series include single, dual and quad-core products. But the "Core" or "Core 2" name itself doesn't tell you that, it's the "duo" or "quad" suffix that denotes the actual number of cores on the chip.
I've always thought that "Core 2" was one of the stupidest names Intel has ever come up with because there's so much potential for confusion.