The twin paradox
A paradox is a seemingly self-contradictory result that presents a puzzle as to its interpretation. The twin paradox is a famous example. Consider a pair of identical twins. One twin becomes an astronaut, and makes a long trip about the galaxy at 95% of the speed of light, while her sister stays home on Earth. After 64 years earth time the astronaut twin returns; but according to clocks and the log book in the space ship the twin has only been gone 20 years because of time dilation. Her appearance shows that she has aged physiologically by only 25 years while her stay-at-home sister has aged by 64 years.
NASA doesn't currently have the capability to build space ships that achieve such high velocities, but the time dilation effect is observed directly every day in laboratories that study subnuclear particles at high energy.
(A paradoxical aspect of this situation arises if you think about it from the point of the view of the astronaut. She sees her sister moving at a high rate of speed; so when they are reunited, she wonders why her sister has aged more rather than less. The explanation involves physics that we will not discuss here, because we have considered only reference frames that move at constant velocity relative to each other. In order for the astronaut to leave and later return, there must also be some acceleration. In the reference frame of the earth, it is only the space ship that accelerates; but in the reference frame of the astronaut, the entire rest of the universe accelerates. This difference is responsible for the difference in what the ladies experience.)