A spaceship leaves earth for a trip at high relativistic velocity. Aboard is one twin, the other having remained on earth. The ship is moving away from earth at high speed, and returns to earth at high speed, therefore it's time is dilated, so the twin on the ship ages more slowly and should return to earth having aged less than his earthbound counterpart.
However, from the perspective of the ship, it is the earth that is moving away at high velocity, and the earthbound twin should be the one aging more slowly.
The paradox dissolves when the nature of each frame of reference is taken into account. The earth is an inertial frame of reference. The spaceship, since it must undergo several positive and negative accelerations to make the round trip, is a non-inertial frame. Only the viewpoint from the intertial frame is valid. The explanation is far too long and involved to go into here, but you will find one of the best and most detailed explanations in Paul Davies' book "About Time" on pages 62 to 65.