If you check some of the recent threads here, you will find that several of them discusses the twin paradox. For example, I wrote a little about it in "Am I thinking SR correctly?" (my last post in that thread). The most recent one is "SR paradox".
The key to understanding the twin "paradox" (which isn't really a paradox of course) is to realize two things:
1. Different observers don't agree on what events are simultaneous.
2. When the twin in the spaceship is on his way back to Earth he's in another frame than he was when he was moving away from Earth.
In the frame moving away from Earth, the twin on Earth is aging more slowly. In the frame going back to Earth, the twin on Earth is...still aging more slowly, but now (in the astronaout twin's frame) he's much older!
It's hard to explain this without spacetime diagrams, but I should at least mention a few basic facts. Every observer thinks of a certain set of 3-dimensional "slices" of spacetime as "space, at different times". But different observers are "slicing" spacetime in different ways.
You can't think of the twin moving away from Earth and the twin going back to Earth as "an observer". He's in a different inertial system on the way back. Because of this, he is now "slicing" spacetime in a different way. What this really means is that events that were simultanous to him when he was moving away from Earth are not simultaneous to him when he's going back to Earth.
The moment before he turned the spaceship around, a certain set of events on Earth were simultaneous (in his frame) with the events he was experiencing. Those are the events that are located on the same "slice" as the "press the return-to-earth button" event. One of those events involves a rather young twin on earth.
When the spaceship has reversed its course (even if it does so very quickly), a completely different set of events are simultaneous with the events he's experiencing. One of them involves a rather old twin on earth.
In other words, they both see each other aging more slowly through the whole trip, except at the turning point, when there's a huge jump in the age difference, from the astronaut's point of wiew. This is not a time dilation effect, but a consequence of the two things I mentioned above.
If you don't understand the stuff I said about "slices", I can only recommend a good introductory textbook on special relativitity, such as "A first course in general relativity" by Bernard Schutz. (I know the title says general relativity, but the chapter about special relativity is still the best introduction to SR I've read). Try to understand spacetime diagrams.
Don't let anyone tell you that you need general relativity here ("because one of the twins is accelerating"). That claim is wrong.