Why is it important that the equation of state parameter of dark energy is measured? The equation of state for a perfect fluid is that $p=\omega \rho c^{2}$, where $p$ is the pressure, $\rho$ is the density, $c$ is the vacuum speed of light, and $\omega$ is called the equation of state parameter. $\omega$ may be constant or varying in time. I'm looking for broad answers (and references if possible) to the following:


*

*Why is it important that the equation of state parameter of dark energy is measured?

*What will it tell us? 

*What are the implications?

 A: It gives us a clue about the future of the universe, since dark energy is the dominant form of energy in the universe and will become even more dominant in the future.  The precise value of $w$ will therefore have a significant effect on the future expansion profile.

This is from a blog post I wrote on the subject, Dark Energy and the Cosmic Horizon.  The value of $w$ has a significant effect on the behavior of the cosmic horizon in the far future - for $w = -1$ it remains a constant size, for $w > -1$ it grows without bound over time, and for $w < -1$ it shrinks.  In the latter case, the scale factor goes to infinity in a finite time - a scenario called the Big Rip.
More generally, measuring dark energy in all the ways we can think of should give us clues about what the heck the stuff is.  Don't forget that "dark" just means "we know it's there, but don't ask us to explain it".  The fact that it's the dominant form of energy in the universe and we have no idea what it is is plenty reason to be curious!
A: References: 
http://wfirst.gsfc.nasa.gov/science/DETF_Report.pdf
http://arxiv.org/abs/1211.0310 
http://sci.esa.int/science-e/www/area/index.cfm?fareaid=102 
You asked for good references. Here are three.  I don’t know your level of sophistication, but these are written for a funder’s point of view and are hence easier to understand.  But the many references contained therein will give you a more detailed science approach if that is what you are interested in.
The first reference, from around 2005, is still regarded as the definitive statement of the problem from a US point of view.  The second, from 2012, is a very good update from one specific project perspective (LSST).  The third, a website, gives you the current take from the leading European related space project (Euclid).
You also asked why this is important.  Everyone seems to agree that the dark energy is the most massive unsolved problem in cosmology.  Let me quote from the top of the third reference:
“Theme: How did the Universe originate and what is it made of?”
Is that important enough for you?  (Grin)  
A: It's Einstein vs. Bohr all over again.  If w is exactly -1 and doesn't change, then Einstein's cosmological constant is the correct explanation.  If it's anything else, then there is some other explanation, most likely a quantum field.  Google or wiki "Quintessence" for one of the leading contenders.  This time, so far, Einstein seems to be winning, but the quantum gang is not yet ready to accept this. 
A: @NathanReed gives a good explanation of some implications. I'll try to better address the other aspects.
Like you've mentioned in the question $\omega$ is some parameter to effectively characterize and model the thermodynamic properties of some stuff (dark energy in this case). It doesn't tell you what that stuff  is actually made up of. We have many different models for what microscopic theory explains dark energy, and they predict different values for $\omega$. So measuring $\omega$ gives us a way to test those microscopic theories. For eg: If dark energy is described by a cosmological constant, then $\omega = -1$. If instead it consisted of (say) scalar or tensor fields, or some combination, then depending on the microscopic details of the theory, different values of $\omega$ will be predicted for dark energy.
Effective combined $omega$ for all the stuff in the universe (together) is a weighted combination of contributions from things like slow moving matter (both baryonic and dark) with $\omega=0$, ultrarelativistic matter with $\omega = \frac{1}{3}$ and dark energy (presumably $\omega \sim -1$). So an accurate measurement of $\omega$ will also give us a clue to the relative abundances of different kinds of stuff.
References


*

*https://en.wikipedia.org/wiki/Equation_of_state_%28cosmology%29

*http://www.scholarpedia.org/article/Dark_energy and further references listed there.

*Standard textbooks on large-scale cosmology shoud talk about this. You could look at (say) Weinberg's text.

