Well it's not that complicated really! The twin paradox is a thought experiment meant to demonstrate Einstein's time diletation of travelling clocks (time progresses slower in traveling clocks).
Lets say we take two twins, Twin E (for Earthling) and Twin A (for Astronaut), keep E on Earth and send A into space. A would have to be in space for a very long time and travel at a high speed. On his return home, A would discover that his twin brother had aged much faster.
How does this work? In E's reference frame, the turning point is 3 light-years away and A travels at 60% of lightspeed (lightspeed c=299,792,458 meters per second) both ways. Thus, the trip takes 5 years each way and home arrival is 10 years after departure from E's point of view. The total travel time t=2L/v (t=time; v=velocity, 0.6c in our case; L=distance traveled away from earth=3 lightyears) = 10 years.
Because of the high speed involved we need to introduce ɣ=(1-v²/c²)^-1/2 (ɣ=gamma=Lorentz-factor) for A. From A's point of view the time he needs is t=2L/ɣv= 8 years!
As you can see the twin who stayed home is now two years older than his astronaut brother.
In the Drawing E (blue) is in the (ct,x)-system and A is in the (ct',x')-system. You can see that as time progresses, and A (red) gets further away from earth, events happen later in his system. This is how the twin paradox works graphically.
So what's so paradox about it? When Einstein postulated his work on relativity and Paul Langevin created this thought experiment, skeptics argued that the twin A could just as well have viewed himself as stationary, with the Earth and his twin brother speeding away from him and return to him later. In such a case, the Earth and twin E must be the younger ones, because they were in the 'moving frame of reference'. When compared to the original postulate, this is obviously paradoxical.
Of course this is not realistic, as twin A has to suffer g-forces to get up to speed, then again to turn around and head back and finally again to land on Earth. So the situation is not symmetrical.