Dual core is a generic term representing 2 cores on a single die. Likewise, quad core represents 4 cores on a single die.
This is different from Core Duo and Core 2 Duo. These are micro-architectures for how the CPU operates. Core was Intel's first dual core architecture, and Core 2 is the "sequel" to Core.
So you see, a CPU can be a Core Duo with 2 cores, or it can be a Core 2 Duo with 2 cores. They are both dual core -- the difference is the micro-architecture.
Simply put, Core 2 is a faster, more powerful micro-architecture.