Which frame of reference to use when calculating time dilation? First of all, I'd like like to apologise. I'm not a physicist so it's gonna be an uphill struggle with explaining some concepts to me ;) 
I think I understand the math behind time dilation between two frames of reference. I have no problem grasping which frame of reference "ages more slowly". I seem to fail at understanding something quite more basic.
What I'm not sure is how to find out "which way" (so to speak) to look at time dilation when speaking about the change of the perception of time.
Consider these factors:
- person A is on Earth (stationary)
- person B - is an astronaut traveling on a rocket at a significant relativistic speed (say 80% of the speed of light) 
- the distance traveled by the rocket equals to 1 light year for the stationary observer
From this I gather that the flight to cover a distance of 1 LY at 80% speed of light would take 1.25 years. This would mean the astronaut travels for 1.25 years.
But according to whom?
My question is, who perceives the time calculated as above, meaning: distance between two objects in space over velocity of the rocket? 


*

*Is '1.25 years' the time perceived by the astronaut (person B) that is then subject to time dilation (lengthening) for person A? In which case:


*

*for person A the time of person B's travel is >1.25 years (the size of dilation is a secondary issue), and

*for person B the time of his own travel =1.25 years.



or is it the other way around?


*Is '1.25 years' the time perceived by the stationary observer (person A) that is then subject to shortening for the astronaut (person B)? In which case:


*

*for person A the time of person B's travel is =1.25 years and 

*for person B the time of his own travel is <1.25 years.



I'm guessing that the answer to my question could be answered by figuring out "according to whom does the rocket (and person B) travel at velocity = 0.8 lightspeed?" 
So, by extension, if a scientist claims to have created a rocket capable of reaching 0.8 lightspeed - does this mean that stationary observers will be able to see it traveling at 0.8 lightspeed or only the crew of the ship will be able to perceive that top speed, while for stationary observers the ship would be traveling at a slower rate (hence the dilation)? Does length contraction play any part in this whole conundrum?
 A: 
From this I gather that the flight to cover a distance of 1 LY at 80%
  speed of light would take 1.25 years. This would mean the astronaut
  travels for 1.25 years.
But according to whom?

According to clocks for which the spacecraft has a relative velocity of 0.8 c.  This isn't even a special relativistic calculation.
If the spacecraft is observed by Alice to have a uniform velocity of 0.8 c, this means that the spacecraft travels a distance of 1 light-year in a time of 1.25 years as measured by Alice's rods and clocks.
Likewise, Bob, the astronaut on the spacecraft, observes that Alice has a uniform velocity of 0.8 c so that Alice travels a distance of 1 light-year in a time of 1.25 years as measured by Bob's rods and clocks.
But, and this where special relativity comes in, Alice and Bob observe each other's clocks to run slower than their own clocks (time dilation).
So, when Alice's clocks show an elapsed time of 1.25 years, Alice observes that Bob's clocks show an elapsed time of 
$$\tau_{B} = \sqrt{1 - 0.8^2}(1.25)\;\mathrm{yr} = 0.75\, \mathrm{yr}$$
Likewise, when Bob's clocks show an elapsed time of 1.25 years, Bob observes Alice's clocks to show an elapsed time of $\tau_{A} = 0.75\;\mathrm{yr}$.
That both observe the other's clocks to run slow may seem paradoxical but, as you look further into it, you'll find that the resolution lies with the fact that Alice and Bob also observe that the other's clocks are not synchronized with each other.
A: I don't think that is a basic question, it's a natural way of thinking, since time dilation and all other relativistic effects aren't present in everyday physics (Newtonian). To simply answer your question, the '1.25 years' time is measured from the spaceship (the clock at rest) and is called the "proper time", while the time measured by person A is called relativistic time and is greater than the proper time, in all cases. So basically, whenever an object moves, the time that is measured from that object's reference frame is the shortest time, while all other times measured from other reference frames are greater.
