Most computers used to have one 'CPU' inside them, that did all the processing work, and was the brains of the PC.
For years, manufacturers like Intel could just add more stuff into the CPU and make it run faster --- from 1 GHz to 2 GHz etc., so that everything magically runs faster.
A few years ago, technical issues (not reaching the entire chip in one clock cycle and heat issues), stopped this magic from being possible as easily. So, manufacturers like Intel started putting two 'cores' on one chip.
So instead of having one brain on one physical microchip, they now have two brains on the one chip. This is called dual core.
If you put four brains on one chip, this is quad core. And you can go up and up, in time.
But this doesn't automatically make things faster.
It takes nine months for a woman to have a baby. But you can't knock up nine women and have a baby in one month. Some problems naturally can be 'parallelised', and some can't.
The problem is that all computer programming techniques have been based with one brain at a time. Changing programs to work with more than one is a lot of work.
Currently, Mac OS X, Windows, Linux, etc., all work well with one core, and in some ways can take advantage of more than one core. But not all that well.
So really, you won't notice much of a difference now between dual core and quad core.
But in a few years, if you upgrade the software, you might. So if the choice is between quad core and dual core, go for quad core.