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In relativity, you have two twins on the Earth. One gets into a space ship and the other stays on the Earth. The one in the space ship then travels at a very close speed compared to the speed of light. Since the twin in the space ship is traveling at a high speed he will experience time dilation. He will not notice any difference in the slowing of time, but his twin on the Earth will he the space ship twin's clock ticking slower than the clocks on Earth and so the twin in the space craft will be experiencing less time relative to the twin on the Earth. Later on the twin in the space ship returns to Earth and he notices that he is younger than his twin on Earth.
I also wanted to add.
Time dilation can be applied only by observers who are in an inertial reference frame (non-accelerating frame). The twin in the space craft can not apply time dilation to his twin on Earth and say that the Earth is moving away so its clock is slow since the space craft is in an accelerating frame.
What was described above is the base situation, it is NOT the paradox itself.
The paradox lies in the fact that from the second twin's point of view, the situation seems symmetrical (first twin seems to be going away and come back). So the travelling twin would feel reasonable to think that his brother would be the younger on when they meet again.
The paradox is lifted by explaining that the situations are not symmetrical, because the travelling twin experiences accelerations at different points of his journey, whereas his brother does not.