While not as spectacular as a solar eclipse, a lunar eclipse can still be a beautiful and amazing spectacle. It's also a lot easier to see a total lunar eclipse than its solar equivalent!
A lunar eclipse always occurs at night, during a Full Moon; you should be able to see the eclipse if it occurs during your nighttime, and you have a view of the Moon. But what you will see depends on the specific type of the eclipse.
By the way, since a lunar eclipse occurs at night, when the Sun isn't around, it's always safe to look at a lunar eclipse.
As described in Mechanics of Lunar Eclipses, the Earth's shadow during a lunar eclipse has two parts: the penumbra, where the Sun's light is only partly blocked from the Moon; and the umbra, where the Earth totally blocks the Sun's light.
If the Moon touches the penumbra, a penumbral eclipse will be seen (although, in practice, this type of eclipse is subtle; most penumbral eclipses can't bee seen with the naked eye); if the Moon touches the umbra, a partial umbral eclipse will be seen; and if the Moon completely enters the umbra, the result is a total umbral eclipse, or total lunar eclipse.
A total eclipse is always preceded and followed by a partial eclipse; which is in turn preceded and followed by a penumbral eclipse. Hence, we say that a total eclipse goes through a penumbral stage, then a partial stage, then the total stage, then partial, then penumbral, before finishing.
The penumbral stage of a lunar eclipse (or a lunar eclipse that is only penumbral) is essentially impossible to see, unless the Moon is deeply inside the penumbral shadow; even then, all parts of the Moon are still receiving light from the Sun, and therefore quite bright. Still, in a deep penumbral eclipse, sharp-eyed observers should see a subtle but distinct shading across the Moon at maximum eclipse. This should be fairly obvious in a total penumbral eclipse.
A partial eclipse, or the partial stage of a total eclipse, is heralded by the Earth's shadow beginning to take a bite out of one side of the Moon. The remainder of the Moon will still be quite bright, as normal; but the Earth's shadow will gradually move across the face of the Moon, and the light level will drop noticeably.
When the Earth's umbral shadow completely covers the Moon, the drama begins! The Moon darkens, but rather than vanishing totally, it turns a dark red or coppery-red colour. This is an amazing sight, and is caused by the refraction of sunlight through the Earth's atmosphere; although the Sun is totally blocked from the Moon, this scattered light, filtered by the atmosphere, ensures that a total lunar eclipse is a sight worth seeing.
This amazing image, by Michele Whitlow, illustrates the stages of the total lunar eclipse of 8 Oct, 2014, from the beginning of the partial eclipse, through the total eclipse, to the end:
Depending on the prevailing condition of the Earth's atmosphere, in terms of cloud cover and dust from volcanic eruptions, the actual colour of the Moon at totality can vary from near black (particularly at mid-totality), rust, brick red, and bright copper-red or orange.
The Moon is quite visibly larger when it's close to the horizon, as in when it's rising or setting — how does this affect an eclipse?
Well, things are not quite what they seem — in fact, this is just an optical illusion. You can prove this by taking a picture of the Moon near the horizon, and another when it's high in the sky; with the same lens settings, the two Moon pictures will be the same size. The Bad Astronomy web site has an article on this subtle and interesting illusion.
So where do you have to go to see a lunar eclipse? The good news is that this isn't too hard.
A lunar eclipse is much easier to see than a solar eclipse. When the Moon moves into the Earth's shadow, half of the Earth — the night side, where the Moon is above the horizon — can observe the event. The type of eclipse you will see doesn't depend on where on Earth you are; as long as you can see the Moon, you will see the same view of what's happening to it at any given moment as everyone else.
Your ability to see the Moon, of course, is governed by the Earth's rotation; as the Earth spins, the Moon, as seen from any given point, appears to rise in the East, move across the sky, and set in the West. Since a lunar eclipse can last for hours, you may find that from your location, the Moon rises or sets part way through, and you only see part of the eclipse; or you may see it all, or none of it.
As an illustration of this, the following map shows the total area of the Earth which witnessed the total lunar eclipse on July 16, 2000:
The orange and yellow shaded areas on the map will see only the beginning of the total eclipse, as the Moon sets in the West; the cyan and blue shaded areas will see just the end, as the Moon rises; and the light area (covering Australia and east Asia in this case) will see the entire spectacle. The green tag in the centre shows where the Moon was directly overhead at the moment of maximum eclipse.
Bear in mind that if you're very near the edge of the area of visibility, the Moon will be low on the horizon; depending on your view of the horizon, you might not actually be able to see it.
The other good news is that a lunar eclipse, even a total eclipse, generally goes at a pretty relaxed pace (the total phase often lasts an hour or more); with that, and the lack of any need for eye protection, there's plenty of time to enjoy the view, take pictures, and have a good time!
And yes, that's right — a lunar eclipse can't damage your eyes. The Moon is no brighter during a lunar eclipse than at any other time, and nowhere near bright enough to cause blindness. (If you're looking at the Sun, that's not a lunar eclipse — and it is dangerous.)