Which may look complicated, but we can break it down to this:
Time elapsed for person at rest = Time elapsed for person in motion
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the square root of
( 1 - speed of traveler squared / speed of light squared)
Let's take an astronaut. The average astronaut in orbit let's say has an orbital velocity of about 5 miles per second. The speed of light is 186000 mi/sec. So we take the square root of one minus the ratios of those two (0.999999999) and we get about the sam4e number (since it is almost 1, this makes sense). The reciprocal of this is 1.00000000004. So we multiply this value times the time traveled by the person in motion. Let's say it is a 10 day journey, or 864000 seconds. That gives us 864000.0003 seconds. That gives us a difference of 0.0003 seconds, which means that the astronaut is 0.0003 seconds younger (or traveled through time) than a stationary person on the earth. That is pretty insignificant timewise, but if we could reach speeds of 90% the speed of light, then a trip that takes 10 days for the astronaut will be observed on earth as taking almost 23 days. That gives us a significant difference of 13 days. Put in terms that may make more sense, let's say the astronaut leaves on Jan 1. When she comes back, she thinks it's 10 days later, Jan 11th. On earth we think it's 23 days later, Jan 24th. Fairly significant time difference.
