Special relativity tells us that a moving clock runs slow. So an astronaut who goes to a distant star at relativistic speeds and comes back will have aged less than his/her twin that stays home.
The apparent paradox lies in the naive application of the principal of relativity to the problem. If I think your clock runs slow because you are moving relative to me, you think my clock runs slow because I am moving relative to you. So you might argue that from the astronauts perspective, the twin at home should age slower.
This is incorrect, however. The paradox is resolved by the fact that the symmetry is broken when the astronaut turns around to come home. His reference frame changes. During that switch, the twin at home appears to him to age a bunch (by application of the Lorentz transformation--of course he doesn't just suddenly "see" the twin age). A good way of seeing exactly what happens is to imagine the twins sending radio messages to each other at fixed intervals. On the way there, the radio signals are redshifted--the interval is increased, so they come less often than they are sent. On the way home, they are blueshifted--the interval is decreased so they come more often. The travelling twin spends an equal time in redshift and blueshift mode. But the home twin sees a long period of redshift and then a shorter blueshift, so he sees fewer signals come from the astronaut, so he knows the astronaut aged less.
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