I can't really tell you a whole heck of a lot about turbulence models because, quite honestly, I hate the core idea behind turbulence modeling so I don't even bother.
As far as the difference between RANS, LES and DNS, it really comes down to trade offs between how much physics is captures and how long the computations take.
A DNS solves the Navier-Stokes equations directly (or any other set of equations for that matter). Because of that, they are capable of capturing pretty much 100% of the physics in the flow and are limited only be computational power and the assumptions you make in setting up the simulations. That is why they are sometimes called numerical experiments. Turbulent flows, as you know, have a variety of scales ranging from the inertial scales down to Kolmogorov scales. Unfortunately, that means that the mesh for a DNS must be incredibly dense to capture all that physical content, so the time to converge on a solution is extraordinarily large. It scales approximately with Re3.
RANS averages the Navier-Stokes equations in order to simplify the equations and make them less computationally intensive to solve. It averages out a lot of the smaller scales, which are the ones that drive up the computational time for the most part. This means you don't need nearly as fine a mesh. That is nice when you don't need the fine detail and a turbulence model like k-ε will do for you just fine. It does mean that you lose a lot of the finer physics of the flow, though. Generally, even complex problems at high Reynolds numbers can be solved in fairly short amounts of time though.
LES falls between the two and is closer in physical accuracy to a DNS than to RANS. It is slightly faster than a DNS and captures slightly less physics. It captures much smaller scales than RANS does, but not nearly those of a DNS. The equations themselves differ from those of RANS but are still not the full Navier-Stokes equations.
There is also a new technique called the partially-averaged Navier-Stokes equations, or PANS. That uses a constant whose value can be used to set the fidelity of the simulation anywhere between that of RANS and LES.
Unfortunately though, I am not a turbulence guy. I work in the area of boundary-layer stability and transition, not turbulence modeling, so I don't know a whole lot about RANS, PANS or LES beyond what I have said here. Certainly not enough that I could take the place of a few journal papers.