Definition
Imagine twins born on the same day. One rides a high-speed train and the other stands on the station platform. Their clocks tick differently. The twin paradox is a thought experiment about what happens when one twin travels through space at nearly the speed of light and returns: he has aged much less than the twin who stayed on Earth. According to relativity, time runs slowly for anything that moves fast. But each twin sees the other as the one who is moving, so the puzzle begins with a question: why is only one of them younger?
Why does each twin claim the other's time runs slower?
When you ride a train and look out the window, the scenery seems to slide backward. To a person standing on the platform, the train is the thing that moves. To a person inside the train, the platform is the thing that pulls away.
Special relativity says that time runs slower for something moving fast, as seen from outside. So the older brother, flying away in a spaceship at enormous speed, would see his younger brother's time on Earth running slowly.
From the point of view of the younger brother staying on Earth, it works the other way around: the time of his brother, drifting away in the spaceship, should be the one running slower. In the end, both claim that the other one's time runs slower. Both claims seem to be correct at the same time, which looks like a contradiction. This situation is called the twin paradox.
To be more precise: the U-turn that breaks the symmetry
If the two twins were in exactly the same situation, this would be a real, unsolvable contradiction. But there is one decisive difference hidden between them. Only the brother in the spaceship reached his destination, stopped, turned around, and went through a U-turn on the way back to Earth.
The brother who stayed on Earth kept the same state of motion from start to finish, with no sharp change in speed or direction. The brother in the spaceship, on the other hand, had to hit the brakes, swing the ship around, and speed up again to get home.
In physics, a change in speed or direction like this is called accelerated motion. At the moment the brother turns around, the Earth time that he counts as "now" jumps far ahead. What creates the time difference is not how strong the acceleration is. It is the asymmetry that only the brother switches to a different frame of reference on the way out and on the way back.
What actually happens when they meet again?
When the spaceship lands safely and the two stand face to face again, there is no more confusion. The brother whose state of motion changed by turning around really does end up much younger than the brother who stayed on Earth.
Suppose the brother flew at a speed very close to the speed of light and traveled for 10 years before returning. The result is startling and depends on his speed. His clock and his body have aged only 10 years, but on Earth 50 or even 70 years have passed, and the brother who stayed behind is a white-haired old man.
This is not a trick from a science-fiction movie. Time dilation has been confirmed again and again, for example by experiments that put precise atomic clocks on airplanes and satellites. We just don't notice it in daily life, but there is no absolute time that flows the same way for everyone.
Common misconceptions
The twin paradox is a real physical contradiction that proves relativity is wrong.
It only looks like a contradiction. The one who turns around and comes back is only the person in the spaceship, so the two twins are not in a symmetric situation, and the result can be explained logically.
Where you meet it
In the movie Interstellar, the daughter has aged far more than the hero who returns. It looks similar, but it is mostly caused by the strong gravity near a black hole, so the cause differs from the speed effect in the twin paradox.
The clocks on GPS satellites, which race around the Earth, run about 38 microseconds per day faster than clocks on the ground because of the differences in speed and gravity. So satellite clocks are set in advance, before launch, to run that much slower.
In one sentence
A person who travels at nearly the speed of light and then turns around and comes back experiences less time than a person who stayed on Earth, and so ends up younger.