What does the first postulate of Special Relativity mean? Physical laws are invariant regardless. It can be an inertial frame and also a non inertial frame. Now it is the case you cannot do any experiment in an inertial frame of reference to test if you are in motion but that is a seperate topic.
Even when you have an accelerating frame of reference and test to see that you are moving the laws of physics are still invariant. It just so happens that you are moving. You are still relying on the invariant nature the laws otherwise you would not be able to test if you are moving or not. I am wondering why Einstein did not just say all frames of reference and leave out inertial. Was it perhaps because he was addressing inertial frames in this theory?
 A: One has to define what physics is and  what a "law" is in physics, first:
Physics is the scientific discipline that studies observations and measurements in nature, and looks for mathematical models that will  fit data and observations and also , this is important, be predictive of new data and observations. ( Otherwise, if not predictive it is a just a map of the data and observations). Physics uses mathematics as a tool, and the models are called theories.
A law, principle,postulate in a theory of physics is an axiomatic statement, which picks up those solutions of the mathematical model that are predictive of new data. If the predictions are wrong  the theory falsified and a new  or improved model should be found.
So it is using the axiomatic laws of special relativity and the axiomatic law of the speed of light that allows special relativity to be predictive in all frames. (General relativity  has different "axioms" , its principles; at the limit of low masses/energies it reduces to special relativity).
And of course postulates, principles and laws depend on the definition of the terms used .
Inertial frame is defined as :" An inertial frame of reference is a reference frame in which a body at rest remains at rest and a body in motion moves at a constant speed in a straight line unless acted on by an outside force."
So  you must see that "Physical laws are invariant regardless." of frame of reference cannot be correct if the laws depend on the frame of reference? 
An accelerating frame makes a body at rest move, after all, so it is not the same as an inertial frame. Extra functions are needed to describe its motion not given by the laws within the frame. The laws are frame dependent by construction .
It is the definition  you give to the word "law" that is different from the way the physics theories are using them.  A law/axiom that declares : "The laws of physics take the same form in all inertial frames of reference"  cannot be extended to non inertial frames if the theory is correct. And special relativity has not been invalidated by any of the experiments that could invalidate it.
A: The first postulate is a statement of the principle of relativity, as first articulated by Galileo.
There is a sense in which physics is invariant with respect to arbitrarily reference bodies. At worst, you're just using different surface forms of what are fundamentally the same physical principles. There's only one objective reality, and noninertial objects occupy it too. However, there's no real physical content in this fact, beyond the basic idea that science describes a real world. Mostly, it's just a statement about the internal consistency of mathematics. The Galilean principle of relativity is an actual strong constraint on what can be determined experimentally and what the physical laws can be. It's a surprising and nonobvious fact about our world that linear motion relative to the fixed stars can't be detected locally in a lab (without looking out the windows), but rotational motion relative to the fixed stars can be. There's an interesting section about this in the Feynman Lectures (I-16-1, online here).
Since this came from HSM, it's worth saying that you could derive special relativity in many ways from different postulates, but Einstein's postulates made sense given his audience at the time, because they concisely summarized the quandary that physicists faced at that time. On the one hand, no one could demonstrate an explicit violation of the principle of relativity. On the other hand, the measured speed of light seemed to be independent of everything else. There's rhetorical appeal in saying, in effect, "instead of trying to reconcile these facts with a preexisting model of the world, let's just assume they're true (since apparently they are) and see where that leads us."
(The extent to which he actually derived special relativity from those postulates is debatable. At the very least he also depended on a large amount of physical intuition – though the same is true of Euclid's postulates and proofs.)
As another historical note, Einstein later introduced a "general principle of relativity" that is supposed to generalize the Galilean principle to arbitrary frames of reference. It has largely fallen by the wayside for the reason I gave above: unlike the Galilean principle, it doesn't really have any physical content. Maybe in the future when we understand how it came to be that rotational motion is detectable and linear motion isn't, it'll return as a description of an underlying symmetry of physics that is broken in our cosmic neighborhood. But we're not there yet.
