The Path Of Totality is the region where the shadow reaches, you must be in this region to see the eclipse. The eclipses rarely last for more than a couplle minutes. As the moon covers the sun there is a halo around the moon called the solar corona, which is plasma heated two million degrees in temperature. It is even more rare to see the corona than it is to see an eclipse itself.
The most recent solar eclipse was August 11, 1999 which you could see from Europe and the Middle East. But sometimes the moon is too small to cover the whole sun causing a partial eclipse of the sun. This happens when the moon covers the sun at the far side of it's orbit making it look smaller.
When the umbral shadow is not large enough to reach Earth it is called antumbral shadow. The path of the antumbral shadow is called the path of annularity. If you are in the area you will see the ring of annularity. These types of eclipses are also very dangerous to look at with the naked eye.