What does the statement "the laws of physics are invariant" mean? In the first paragraph of Wikipedia's article on special relativity, it states one of the assumptions of special relativity is

the laws of physics are invariant (i.e., identical) in all inertial systems (non-accelerating frames of reference)

What does this mean? I have seen this phrase several times, but it seems very vague. Unlike saying the speed of light is constant, this phrase doesn't specify what laws are invariant or even what it means to be invariant/identical. 
My Question 
Can someone clarify the meaning  of this statement? 
(I obviously know what an inertial frame is)
 A: 
The laws of physics are invariant

means slightly different, but (almost) equivalent things depending on what formulation you are working with.
Given a collection of transformations (a symmetry/transformation group) and a Lagrangian formulation, you can check whether the Lagrangian changes when you apply the transformation. If it does not change (or only by a total derivative), then the action is invariant under the transfomation, and using the principle of extremal action will yield the same equations of motion as before in the sense that they extremalize the same action, and hence describe the exact same system.
Given a collection of transformations and a Hamiltonian formulation, it is of course the Hamiltonian that has to be invariant. The Hamiltonian formalism is not manifestly Lorentz invariant, and it is a bit difficult to use it for relativity, but one can do so. Again, a Hamiltonian unchanging under a transformation induces physically equivalent equations of motion describing the exact same system.
In the case of the statement about inertial frames, the corresponding transformations are given by the Lorentz group $\mathrm{SO}(1,3)$.
A: 
this phrase doesn't specify what laws are invariant

It doesn't need to since it is a guiding principle, a razor.  It is a statement about the nature of physical law.
Put another way, on this principle, an alleged 'physical law' that isn't invariant under inertial coordinate transformations is not a genuine physical law.

or even what it means to be invariant/identical.

Consider, for example
$$\vec F = m \vec a $$
If this equation holds in one coordinate system, it holds in all the coordinate systems related to this one by a Galilean transformation.  Thus, it is invariant (unchanged) by this transformation.
A: The laws of
physics are the same in every inertial frame of reference.
If the laws differed,
that difference could distinguish one inertial frame from the others or make one
frame somehow more correct than another. Here are two examples:
Suppose  you watch two children playing catch with a ball while the three of you are  aboard a train moving with constant velocity. Your observations of the motion of
the ball, no matter how carefully done, cant tell you how fast (or whether) the
train is moving. This is because Newtons laws of motion are the same in every
inertial frame.   
Another example is the electromotive force (emf) induced in a coil of wire by  a nearby moving permanent magnet. In the frame of reference in which the coil is  stationary  the moving magnet causes a change of magnetic flux
through the coil, and this induces an emf. In a different frame of reference in
which the magnet is stationary the motion of the coil through a magnetic field induces the emf. According to the principle of relativity, both of these
frames of reference are equally valid. Hence the same emf must be induced in
both situations.
(Examples are taken from the book ,UNIVERSITY PHYSICS).
A: According to Einstein:

All our well-substantiated space-time propositions [and consequently, all of our statements concerning facts and findings in physics] amount to the determination of space-time coincidences. If, for example, the [course of events] consisted in the motion of material points, then [...] nothing else are really observable except the encounters between two or more of these material points.

where "determination of space-time coincidences" is thought, at least in principle, to be unambiguously, definitively and consistently obtained by each individual participant.
The statement that 

the laws of physics are invariant (i.e., identical) in all inertial systems (non-accelerating frames of reference)

can be understood as a less precise (possibly circular) and more restrictive formulation of Einstein's maxime quoted above. (It is based on Einstein's earliest, preliminary attempts at trying to express his maxime.)  

(I obviously know what an inertial frame is)

Really?!? (cmp. "What determines which frames are inertial frames?", PSE/q/3193)
A: In leyman's terms, it just means that the laws of physics are the same everywhere. Here, on the Moon, even in another galaxy, or in a spaceship  travelling at near light speed to another galaxy.
A: "The laws of physics are invariant in all inertial systems." is a statement. It's something somebody says to express an idea. Its not an object. An acquaintance who works in a restaurant once said "As long as nobody places any orders, we have everything completely under control." which basically says the exact same thing.
It means more than one thing.
It supports the notion that the laws of physics are only theory. As beautiful and absolute as they may sometimes seem, they do not define reality. They are merely a 'user level' interpretation of it. It being quite possible to successfully work with the laws of physics without understanding them. Nevertheless, their frame is by this statement limited to significantly inertial systems, since reality knows no such thing as a perfectly inertial system. The mentioned significance being something that can be calculated. The laws of physics are by this statement only invariably applicable on systems that can be considered absolutely inertial.
One could say the statement alerts to the fact that outside the realm of significantly inertial systems, the laws of physics will fail to give usable results.
It also means that as long as you have a grip on all (significant) dynamic factors in a system, you can count on the laws of physics to give you correct results that express a matching image of the experience of reality, even if you don't fully understand these laws.
It may also tell you, that if you find the laws of physics to apply successfully on a system, meaning any system at all, you're dealing with a (significantly) inertial system, even far outside the realm of physics.
Finally it tells you, that if you find the laws of physics to fail in giving you a matching image of reality and you want to bug-hunt that problem, what you need to look for is an unknown dynamic factor to add to your formulas and thereby to your image of reality.
Any law this does not apply on, is by this statement not to be interpreted as a "Law of Physics." The beauty of it is, that 'true' laws of physics can by the same precondition be successfully applied in fields other than physics.
A: "the laws of physics are invariant..."
means that they wouldn't vary or change.
Meaning that any experiment done in one inertial frame would give the same result as the same experiment done in another inertial frame.
It could be any experiment at all, for example, seeing how the momentum of a ball changed if a given force were applied,  measuring the angle of the maxima in a diffraction experiment, timing the oscillation of a mass-spring system etc...
Every experiment, if set up the same, would give the same result.  Hence two experimenters would deduce the same laws of physics from their experiments.
A: When constructing equations of motion which are the reflection of laws of nature so to speak, we must make them Lorentz invariant and invariant to spacial rotations. This means that they must have the same form under these transformations. One example is construction of a field theory, in which you begin by forming an action which is Lorentz invariant making sure from the very start that you will get it right. Action is a physical quantity with a dimension of Js (joule-second). This quantity is very important for the thing called Hamilton principal of stationary action...So laws of nature same in all inertial reff frames = equations that describe them invariant with form to Lorentz transformations.
A: In layman's terms, it just means that the laws of physics are the same everywhere. This means that we are talking about one common set of laws. The fun part is figuring out how one common set of laws can behave the same, while they are taking place within different frames of reference. Thus we have a one, that is shared by a many. How can this be, when each frame of reference is different. 
Of course once you fully understand both the cause and structure of Special Relativity, the answer becomes obvious.
A: As pointed out by John Hunter the statement means if you done  experiment $A$ in one inertial frame with velocity $v$ and you do the same experiment $A$ in another  inertial frame with velocity $v'$ you will get the same result.
Here we have to be distinguish between realizing an experiment in an inertial and  observing a experiment in an inertial frame.
As example suppose  an inertial frame with velocity $v$ measure the frequency of light emitted by source in his lab frame and find it to be $\nu$. Another frame with velocity with $v'$ with the same source in his lab frame will measure the same frequency $
\nu$.
Now  if they are measuring the frequency of another light source in another frame their measurement will be  different due to the doppler effect.
