In reaction-diffusion processes what is the difference between oscillatory media and excitable media? What is the basic differences between oscillatory media and excitable media? I know that both comes under reaction-diffusion processes. Where do Turing patterns come in the picture? Can some one give me examples too?
 A: First of all, there are no universally accepted defintions of Turing pattern, excitable or oscillatory medium. This is somewhat understandable as there is no real need for one. All these terms describe very huge classes of phenomena and thus you do not form any statements requiring a precise definitions such as: “If X is an oscillatory medium, then …” That being said:
Excitable medium
From Zykov – Scholarpedia: Excitable media:

An excitable medium is a dynamical system distributed continuously in space, each elementary segment of which possesses the property of excitability.

Excitablity is generally understood as the property of a system (here, an elementary segment) to yield a huge response to a small perturbation and return back to the previous behaviour afterwards.
A classical example for a excitable system is a patch of forest. You can excite it by tossing a burning match into it, the excitation being a forest fire. The system eventually returns to its previous state when the forest regrows. The corresponding excitable medium would be an entire forest.
Oscillatory medium
From Pikovsky, Rosenblum, Kurths – Synchronization, p. 266:

An oscillatory medium is an extended system, where each site (element) performs self-sustained oscillations.

As an example, they give is a container in which the Belousov–Zhabotinsky reaction happens.

Both, excitable and oscillatory media can exhibit spatiotemporal patterns, the former as reaction to an intial excitation, the latter by themselves. In both cases, we need the sites (patches, segments) of the system to communicate with each other either to propagate the excitation or to synchronise the oscillations to some extent.
You can also have media that are both excitable and oscillatory. Here the unexcited state is not a fixed point, but the oscillation. For an example consider an ecosystem where the populations of individual species oscillate due to biological or external rhythms (such as the year) but which also can be excited by external influences yielding beneficial conditions for one species, whose population then explodes.

I am no expert on Turing patterns, but the main differences to the patterns exhibited by excitable or oscillatory media is that they temporally stable. If my understanding is correct, they are potential stable states of certain reaction–diffusion systems.
