Most (or at least, older), computers have a single processor, and you'll have seen Pentium 1 or Pentium 3 or Pentium 4 on the box.
This is the processor, the brains of the machine that does all the work. It's just one single processor.
The processor has a clock inside it, which lets it time its cycles for doing work. This is the GHz number you see. To get more work done (i.e., work faster for you), you want a higher GHz number. This means, a higher clock speed is required.
However, if you increase the clock speed (the usual way manufacturers of processors tend to make them faster), you also get some bad things too. For one, you increase the heat output of the processor. This is generally bad, especially for confined computers like laptops.
As a solution, imagine you get two of these processors and sandwich them together to put them on one physical board. This is known as dual core. Each processor is a core, and you have two of them on one actual 'chip' or 'board', so it's a dual core.
This happily speeds things up, and actually doesn't have all the bad things associated with it as plain clock speed increasing did.
For the end user, it means that you get better performance.
(To complicate matters, you have to also make sure that the software, the operating system, such as Windows that's running on it, supports dual core. Not all versions of Windows do, but most recent versions do, as far as I know.)
Make sense?