For thousands of years people reacted to eclipses with horror. The Chinese word for eclipse means "eat," since it appeared that the Sun or Moon was being consumed. Lunar eclipses would have been quite an ordeal, lasting for a long time, and the face of the Moon the color of blood when it's "consumed."
Greek historian Thucydides (460-395 B.C.E.) recounted how the Athenian fleet laying siege to Syracuse was lost following a lunar eclipse. The navy was ready to sail home, but a lunar eclipse occurred. This was deemed to be a bad omen for their departure, so they stayed. Syracuse broke the siege and destroyed the navy, leaving Athens vulnerable to its great enemy Sparta.
Famously, Christopher Columbus, stranded on a Caribbean island, used the knowledge of an impending lunar eclipse to provision his crew. He convinced the locals that the gods were angry with them and as a sign of disfavor would take the Moon away. With the Moon in shadow, Columbus offered to intercede in getting it back - in return for food for his crew.
Eclipses happen because of the interactions of the Moon, the Sun and the Earth as the Moon and Earth move in their orbits. Some of the ancient Greek philosophers, including Aristotle (384-382 B.C.E.), realized that the darkening of the Moon during a lunar eclipse was the Earth's shadow. From this they also understood that the Earth was spherical because the shape of the shadow was always round.
A lunar eclipse only occurs at the full moon because that's the only time the Moon could pass through the Earth's shadow. Looking at a two-dimensional diagram, you might wonder why there isn't an eclipse every month. It's because the Moon's orbit is slightly tilted with respect to the Earth's orbit. The two points where its orbit crosses Earth orbit are called nodes. Only if the Moon is full when it's near a node do the Sun, Earth and Moon line up for a lunar eclipse.
A solar eclipse occurs when a new moon is aligned with the Earth and the Sun. Since the Moon is small compared to the Earth, it casts a narrow shadow. Therefore the Sun is never totally eclipsed for more than seven and a half minutes, and the eclipse isn't visible except from a narrow band of Earth. However the Earth is large and comparatively close to the Moon, so it casts a large shadow during a lunar eclipse. During a lunar eclipse totality goes on for around an hour and is visible from the whole night-side of Earth.
There are two parts to the Earth's shadow: the penumbra and the umbra. The umbra is darker because Earth blocks the Sun's direct rays, but there is some direct sunlight in the penumbra. You would scarcely notice the subtle darkening of the Moon passing through the penumbra.
When the Moon is fully eclipsed, it's a reddish color. It would be black if Earth had no atmosphere. However sunlight filters through the atmosphere. Sunlight is a mixture of all the colors of the spectrum and our atmosphere tends to absorb the blue end of the spectrum, leaving the red colors. The more air the light travels through, the redder it becomes. If you look overhead at the blue sky, you will see that the blue color is most intense straight overhead, since the light travels the shortest distance to your eyes. If you're looking at the setting Sun, the light has traveled a much longer distance and is noticeably red.
At totality Earth blocks all direct sunlight, but some light comes the long way through the atmosphere. The eclipse varies from a brownish color to a bright copper red, depending on the amount of dust in the atmosphere. Volcanic dust can produce intensely-colored sunsets, but cause dark eclipses.
Interestingly, if you were on the Moon during a total lunar eclipse, you'd see a solar eclipse. Earth's night-side would be faintly visible, as in this work by Hana Gartstein. The Sun and the Moon appear about the same size from Earth, so the Moon's disk just covers the Sun's disk. This gives us a good view of the Sun's corona. However when the Earth is seen from the Moon, it appears much larger than the Sun, so it would block out most of the corona. The reddish circle is sunlight filtering through Earth's atmosphere.
(1) Fred Espenak, "Lunar Eclipses for Beginners" http://www.mreclipse.com/Special/LEprimer.html
(2) Wil Milan, "Lunar Lore: Eclipses through the Ages" http://www.rense.com/ufo6/lunarlore.htm
You can see related images on my Pinterest board Eclipses.