In what ways can a lunar eclipse occur?
Also, on what percentage of the Earth are they usually viewable?
I am aware that there are multiple configurations that constitute a lunar eclipse (umbral, penumbral, partial) and would like more information about each and how they occur.
A lunar eclipse occurs when the moon passes through the shadow of the Earth. That is the only configuration possible.
That being said there are different type of lunar eclipses, namely total(umbral), partial, and penumbral.
A total lunar eclipse occurs when the full lunar disk passes into the full shadow of the earth, the Umbra and the entire disk darkens. During totality (when the moon is completely inside the shadow), the moon is usually not completely dark but has a reddish cast to it. This is due to light being refracted through the earth's atmosphere into the shadow region and onto the moon. The light is red because the blue light is more easily scattered out of the light by the atmosphere.
A partial eclipse occurs when the moon passes through the edge of the earth's umbra. In this case, only a portion of the moon's surface is in shadow and the thus it is partially eclipsed.
A penumbral eclipse occurs when the moon only passes through the partial shadow of the earth (the penumbra). This is the portion of the earth's shadow where the sun is only partially blocked by the disk of the earth. This results in a dimming of the moon but not total obscuration as there is still sunlight reaching the moon's surface.
Unlike solar eclipses, which can only be seen along certain tracks on the earth's surface, a lunar eclipse can be seen anywhere on the earth where you can see the moon. If the moon is above the horizon, you can observe the eclipse. It might not be in a favorable position (low on the sky, obscured by some local bit of geography, etc.) but it is at least potenitally visible.
Also, the Lunar Eclipse article on wikipedia is very good.
All lunar eclipses occur in essentially the same way: the Moon passes through the shadow of the Earth. The Earth's shadow consists of a dark central umbra, which then shades outward to a barely detectable penumbra. The different kinds of lunar eclipse just depend on the angle at which the Moon passes through the shadow. If it passes very far from the centre, it's a partial. If it passes through the centre of the umbra, it's total.
Most lunar eclipses are visible over slightly more than half the Earth's surface. It's seen by everyone on the side of the Earth where it occurs, plus a little bit more because the Earth rotates during the eclipse.