First Postulate of Special Relativity: What does it mean? Wikipedia has this quote:

Special principle of relativity: If a system of coordinates K is
  chosen so that, in relation to it, physical laws hold good in their
  simplest form, the same laws hold good in relation to any other system
  of coordinates K' moving in uniform translation relatively to K. —
  Albert Einstein: The Foundation of the General Theory of Relativity,
  Part A, §1

Does this simply mean that any sound theory expressed in K, should be able to withstand a transfer to another system Z and still hold "true"? Or is there more to "uniform translation relatively to K"?
 A: 
Does this simply mean that any sound theory expressed in K, should be able to withstand a transfer to another system Z and still hold "true"? Or is there more to "uniform translation relatively to K"?

The 'uniform translation ...' part is crucial. "K' moving in uniform translation relatively to K" means that the relative velocity between them is constant (i.e. the acceleration of K' is zero). In this case the laws in K' will look the same as the laws in K. For example, observers in K and in K' will agree on the equations of motion that describe the system. They will be the same equations of motion.
What happens if the relative velocity is not constant? If K' is accelerating relative to K, the equations of motion that K' observes will not be the same as those of K (unless you are doing general-relativity by adding a dynamical metric, but this is beyond the scope). You will still be able to translate results between K and K', but the laws, the equations of motion etc., will look different.
A: Yes: if a theory is sound then its physical predictions of phenomena will be the same regardless of which frame the analysis is done in. Note, though, that some intermediate stages of the analysis, such as electric and magnetic field, may look different in different frames, but the final result is always the same.
This presupposes, of course, the fact that both frames of reference be inertial and thus move in a straight line with respect to each other, which I believe (and the comments below confirm) is not the OP's problem with the postulate. Any mention of general coordinates is silly in this context.
A: It also means it is impossible to experimentaly distinguish one from the other two referentials in relative uniform translation. 
