The wikipedia page repeatedly says that the twin travelling in space is the only one which travels, and also is the only one which faces acceleration and deceleration. So it does not age, while the twin which remains in one place ages.
However, this seems wrong as there is no such thing as staying in one place. The twin which presumably stays on earth is travelling at the same relative velocity, relative to the twin in the spaceship. Since the spaceship twin's acceleration is just the rate of change of velocity, so when the spaceship twin's velocity changes relative to the earth twin, the earth twin's relative velocity compared to the space twin also changes, and so the earth twin experiences the same relative acceleration as the space twin (in the opposite direction).
In fact, the space twin sees the earth twin as accelerating away from him, then decelerating and coming back, while he thinks his spaceship is at rest all the time. So then why should the earth twin age more than the space twin? Why not the other way around?