Eclipses are a temporal alignment of the 3 bodies mentioned. Because those bodies are in constant relative motion -- and you have to be in the right location relative to that alignment to experience it (at least for solar eclipses) -- not everyone gets to see them.
In particular, to view a solar eclipse, you need to be somewhere "in line" with the line from 1) the center of the earth to 2) the moon to 3) the sun ... Of course, moon's shadow on the earth is not just a point on the earth but a circle, so you don't have to be precisely "in line", but close (within a few dozen miles, but you can do this computation yourself if you know the diameter of the earth, moon and sun). The circular shadow that crosses the surface of the earth moves in time (as the moon revolves around the earth and as the earth moves about the sun) so different places see it at different times. Finally, when the eclipse is over, it means that the moon's shadow has moved off the earth's surface and back out into space (where it is +99% of the time).
For lunar eclipses, you only have to be able to view the moon (at night and during the period of eclipse of course). That means if you can see the moon (about 1/2 the surface of the earth can see the moon at any time) you'll see the eclipse! This is because the entire earth is casting the shadow on the moon and when you can see the moon, by definition, you are on the side of the earth that is casting the shadow and hence in position to see that shadow transit the moon.
Again: all these effects are temporal since the earth/moon system is moving constantly (relative to the sun).
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