The Twin paradox is a thought experiment in special relativity (SR): of two twin brothers, one undertakes a long space journey while the other remains on Earth. When the traveller finally returns to Earth, it is observed that he is younger than the twin who stayed put.
This outcome is predicted by special relativity ("time dilation of moving clocks") and can also be verified experimentally, for example with muons produced in the upper atmosphere being detectable on the ground. Without time dilation, the muons would decay long before reaching the ground.
The paradox arises if one takes the position of the travelling twin: from his perspective, his brother on Earth is moving away quickly, and eventually comes close again. So the traveller can regard his brother on Earth to be a "moving clock" which should experience time dilation. Special relativity says that all observers are equivalent, and no particular frame of reference is privileged. Hence, the travelling twin, upon return to Earth, would expect to find his brother to be younger than himself, contrary to that brother's expectations. Which twin is correct?
It turns out that the travelling twin's expectation is mistaken: special relativity does not say that all observers are equivalent, only that all observers in inertial frames are equivalent, i.e. observers which don't undergo acceleration. But the travelling twin most certainly accelerated at least once during his journey and his is therefore not an inertial frame. The twin on Earth rests in an inertial frame for the whole duration of the flight (if we ignore the comparatively small acceleration resulting from Earth's mass and movement) and he is therefore able to distinguish himself from the travelling twin.
In resolving the paradox, it is sometimes claimed that special relativity cannot be applied to accelerating bodies, and that general relativity has to be used, but this is not correct. For instance, the age of both the Earthbound and travelling twin can be correctly calculated by integrating the spacetime interval (or proper time) over the spacetime paths they make in any inertial frame (these paths are known as the twin's worldlines). Similar methods can be used to calculate the relativistic behaviour of an accelerating spacecraft (see relativistic rocket). SR only becomes inapplicable when the effect of gravity is non-negligible, in which case general relativity must be used.
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