Why time slows only for moving object and not stationary observer? How the "stationary" & "moving" are decided? Explanation for the questions in the title:
According to the the (special theory of) Relativity, if an observer is stationary and sees a fast moving object then time runs faster for the observer compared to the mover.
For example, persons 'A' & 'B' are somewhere far away in universe standing on a platform. There is nothing nearby for several light years. Hypothetically assume that 'A' is standing on the platform and 'B' boards the rocket and flies away with a significant % of light speed. Due to which the time runs 1.67 faster for 'A' compared to 'B'.
When 'B' returns to see 'A' after 10 years, 'A' has already passed 16.7 years. This is the premises of Relativity.

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Now, my confusion starts here. Why only 'A' is considered "stationary" and 'B' as "moving"? Simulate the situation in other way for the same event. With respect to 'B', rocket is stationary and 'A' moves away with platform. And finally 'A' "returns" to see 'B'. In such case, 'B' should have grown older by 1.67 times.
But neither that happens nor both 'A' & 'B' age equally. It's just that 'B' remains younger.
On the funny note, the premises of relativity here should be who travels relative to other and not who burns the fuel! :-)
I have referred few questions in this forum, but couldn't get the answer.
 A: @CuriousOne posted this answer in the comments:

The premise of relativity is that the speed of light is the same for all observers. This has consequences, but it doesn't change time. All clocks still behave exactly the same for all observers traveling with their own clocks. It is only between observers that clocks are running at different rate. This change in relative clock times is one of the consequences of the constancy of the speed of light. So you always know what is at rest (the clock next to you) and what moves (the clock on the rocket). The astronaut has a resting clock next to him and you and your clock are moving for him.

A: As the person B takes off in a rocket, both of them would see the other clock move at a slower rate, assuming the rocket to be moving at a constant speed, both are in an inertial frames of reference, but when B wants to return to A, B should make a turn somewhere, so a turn, is an acceleration, and accelerating frame of reference is non-inertial, and this 'turn' or the acceleration can be detected by B, and ofcourse by A, So by making a turn it is clear to both of them that it was B who was travelling and not A. 
The same thing happens in the second case too, they both agree on who was actually travelling
