A correlation meter (also called a phase meter) shows how different the left and right channels are from one another.
If both sides are exactly the same (mono), you can say that they are fully correlated, having a correlation ratio of 1:1, or a correlation factor of 1. If both sides are completely different, they are said to have a correlation of -1. A correlation meter reads from -1 to 1, from fully uncorrelated to fully correlated.
Real mixes bounce around between 0 and 1, showing that the L-R differences are varying, as you'd expect them to be. Correlation meters have persistence, meaning instantaneous values stay onscreen awhile so you can see the accumulated range of values over the past N milliseconds. The wider the range of values, the wider your mix is going to sound. A mix that sticks over to the -1 side has polarity problems and is not mono-compatible.
A goniometer (sometimes called a vectorscope) is a variation on the same idea, except that it displays a Y axis in addition to the X axis of a correlation meter in order to add amplitude into the information. You use it the same way, except that a nice wide mix will have an oval shape, and a mono mix will show as a vertical line. It also has the advantage of showing L-R balance, although if you have two ears and two speakers you've probably already got that covered.