Shadows
Both solar and lunar eclipses are shadows. In the case of a solar eclipse, observers on Earth see the shadow; during a lunar eclipse, they see the object on which the shadow falls. If you were to witness a solar eclipse from space, you would see a large shadow blotting out much of the Earth's bright side. This shadow moves as the moon moves around the Earth. The eclipse ends when the moon passes out of line between the sun and the Earth. The duration of the eclipse varies for people within the shadow, depending on their location. Someone in the center of the eclipse's shadow would witness a total eclipse that lasts the maximum duration. Those on the edges would witness a partial eclipse for a shorter time. People in different places would thus perceive the duration of the eclipse differently.
Time Zones
The moon's shadow is large enough to span continents, so people on one coast will see the eclipse happen at a different time than those on the other coast due to time zone differences. During lunar eclipses, the same principle applies; the Earth's shadow may fall on the moon at midnight in one time zone, but at 11 a.m. for those watching from one time zone to the west. Because all three bodies remain in motion relative to one another throughout the eclipse, the shadow itself moves through different time zones, making timing even more complex.
Solar Versus Lunar Eclipses
Both solar and lunar eclipses occur a few times per year. However, the moon's shadow covers only a portion of the Earth at once during a solar eclipse, while the Earth's shadow covers the entire moon at once during a lunar eclipse. Everyone on the night side of the Earth can witness a partial or total lunar eclipse as it happens, but only some people on the sunlit side of the planet will witness a total solar eclipse during each eclipse event. Anyone who sees a solar eclipse may wait a lifetime to see another one in the same area. From that standpoint, different places will observe separate solar eclipse events at dramatically different times.
Timing Eclipses
Scientists can predict eclipse locations and durations with a high degree of accuracy, but even early astronomers noticed that lunar eclipses happened on a regular cycle. The Saros cycle lasts slightly longer than 18 years and predicts when lunar eclipses will occur. Saros cycles themselves come in cycles. Computers make keeping track of these celestial motions and long-term cycles simpler, and allow astronomers to time the duration of eclipses to within milliseconds.