A total solar eclipse happens only when the New Moon is close to Earth (casting the largest possible shadow) and passing directly between us and the Sun. People in the umbra experience a total eclipse, and people in the penumbra experience a partial eclipse. The Moon only casts a relatively small shadow on the Earth, so only people in a small region of the globe (where the shadow falls, below) can see a solar eclipse.
Solar eclipses happen quickly, and the moment of totality (when the photosphere of the Sun is completely blocked and the corona is visible around the edges of the moon, as above) is only a few minutes long. Before and after totality, the Sun appears to have increasingly larger and smaller bites taken out of it by the Moon.
During an eclipse, the Sun is not any more dangerous to view than on a normal day, but it is more interesting, and we are more likely to stare at it, causing damage to our eyes. Read eclipse-viewing tips before attempting to watch a partial or total solar eclipse.