Relativity and how speed affects passage of time This is one where it doesn't matter how many books I've read about it, they all seem to evade the elephant in the room - from examples like the mirror on the boat reflecting a single photon up and down to a rocket approaching the speed of light.
The question is: the books talk about no absolute speed or movement ie - if two bodies are floating in space then from both perspectives neither of them can say who is "moving" therefore who has a "speed".  A train leaving a platform can be perceptibly seen to be moving away from he platform - but from the passenger's point of view - is not the platform also moving away from the train?  
On these bases - in the case of the astronaut "approaching the speed of light" - my question is "compared to what"? 
And if there is an answer to this - e.g "the earth" - then why do we insist there is no absolute position in space?  One thing must be moving and the other must be still in order for this to be logical - otherwise they are both moving, relative to each other and the effect of speed should logically cancel out?  or maybe I am missing the effect of "mass".
I am not classically trained in Physics but an keenly interested so for me the popular books seem to very frustratingly gloss over this nuance and make the whole Relativity thing difficult to accept ... 
 A: I guess the essence of your question is in here:

On these bases - in the case of the astronaut "approaching the speed
  of light" - my question is "compared to what"?
And if there is an answer to this - e.g "the earth"...

No, not "the earth".
When someone says "astronaut is approaching speed of light" it is assumed "in my frame of reference". Or at least it should be obvious what frame of reference he is talking about.

The question is: the books talk about no absolute speed or movement ie
  - if two bodies are floating in space then from both perspectives neither of them can say who is "moving" therefore who has a "speed". A
  train leaving a platform can be perceptibly seen to be moving away
  from he platform - but from the passenger's point of view - is not the
  platform also moving away from the train?

I think you are correct here.
We have some process (let it better be "train is moving along the platform"). And we can look at this process from different frames of reference. In the frame of reference of a person who is staying on platform, the platform if at rest, the train is moving. In the frame of reference of a person who is sitting on the train, the train is at rest, the platform is moving.
There is no way to tell, which frame of reference is better, which one is "truly staying still". How would possibly these two persons argue whose frame of reference is better? "Almost everything around me is not moving, so my velocity is really zero!". "Not at all, it's only the earth which is not moving in your frame of reference, there are other, bigger and possibly better planets which are moving, so what?".
Better argument would be something like "Look, when this toy canon fires, it does not matter which way it is directed, the maximum fire distance is always 1 meter! That means I am not moving. Would I be moving, the fire distance would be longer (or shorter?) when I point the canon in the direction of my own velocity!". You see, it's some experiment that one can make and find out if it is moving or not without looking around. But the results of this experiment would be the same on the moving train. It will not help to find out if the train is moving or staying.
And no other experiment would do. This is because physics lows are the same in all the systems of reference. (well, in "inertial" systems of reference. Physics lows are quite different in f.e. rotating systems of reference - that's why it's possible to find out if the room you are in is rotating or not without looking out of the window)
You can choose any inertial frame of reference to describe the world around. Some of them may be more or less convenient for some particular problem, but there are no major differences.
And this is the main point of relativity theory! You can choose any inertial frame of reference, and neither of them is better than another.
This was quite obvious for physicists for couple of hundred years until they found that speed of light does not depend on speed of the source of light. Here it is, the speed of light is that much only in some truly staying frame of reference! One can measure the speed of light and find out if he is moving or not.
And all the special relativity theory is the explanation how can it be that speed of light does not depend on speed of it's source, but still there is no "truly not moving" frame of reference.
