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 several hours with the face of the Moon the color of blood as 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 of 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.
The answer is that 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. 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, as you can see in Fred Espenak's diagram. 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.
There are at least two lunar eclipses a year, but in most of them Earth's shadow covers just part of the Moon. There was a total lunar eclipse on December 21, 2010 and another on June 15, 2011. The one after that will be on December 10, 2011.
Interestingly, if you were on the Moon during a lunar eclipse, you'd see a solar eclipse. Click to see what this might look like according to French astronomer and artist Lucien Rudaux. The painting shows Earth's disk as black, but except for a very dark eclipse, Earth's night-side would be faintly visible, as in this work by Hana Gartstein.
Rudaux's painting shows the Sun's corona around the disk of the Earth, but most of the corona wouldn't be visible. The Sun and the Moon appear to be the same size viewed from Earth, so the the disk of the Moon can just cover the disk of the Sun in a solar eclipse. However, seen from the Moon, the Earth appears four times larger than the Sun, so it would block out most of the corona.
The reddish surface of the Moon and reddish circle around the Earth are striking in Rudaux's painting. You can also see the circle around Earth in Garstein's picture. What causes that?
The Moon would be black in a lunar eclipse 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.
When the Moon is totally eclipsed, 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. Richard Keen and SpaceWeather.com show how eclipse color is affected by volcanic dust.
(1) Fred Espenak, "Lunar Eclipses for Beginners" http://www.mreclipse.com/Special/LEprimer.html
(2) Wil Milan, "Fear and Awe: Eclipses through the Agesī"
(3) Jay Pasachoff, "Eclipses of the sun and moon," Knol (Google.com) [Version 33]