The twin paradox
A paradox is a seemingly self-contradictory result that presents a puzzle as to its interpretation. The twin paradox is a rather famous example. Consider twins separated at birth. One twin is sent traveling at 95% the speed of light around the galaxy. The second twin remains on Earth. After 80 years the astronaut twin returns, but according to clocks in the space ship the twin has only been gone 25 years due to time dilation. Thus one twin is physiologically 25 years old while the other is 80 years old.
The reason this provides a paradox is that to the twin in the rocket, the first Earth-bound twin's clocks should be slow, and should therefore be slower. Both twins can not be younger than the other twin. However, the relations for time dilation were derived for clocks in uniform reference frame. The rather complex considerations of accelerating objects would shown that indeed, the astronaut twin has not aged nearly so much as his Earth-bound duplicate.