When referring to computer processors, dual-core is a technology that enables two complete processing units (cores) to run in parallel on a single chip. This gives the user virtually twice as much power in a single chip. For the computer to take full advantage of dual-core, it must be running on an operating system that supports TLP and for applications to support TLP.
You can think of a computer with dual-core as a computer that has two CPUs (processors). If the computer is a quad-core, you could think of that computer as having four processors.