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I have good background knowledge on Quantum Chemistry calculations but my knowledge on Solvation Models is not that amazing.

Many times I read a paper that uses TD-DFT, they also assign some solvent to their calculations. Some papers use Dielectric-PCM and some others use Capacitive-PCM. I know that D-PCM assumes the solvent as Dielectric media and C-PCM assumes as Capacitive Field and I have some very basic knowledge on them but what I can not understand is when we use which?

For example what is the criteria for selecting DPCM or CPCM for Methanol solvent?

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Short answer: It matters most that you are using a modern solvation model than which one

Slightly longer answer: If you have the time, try a few solvation models, compare with experiment and see which gives you the best results. In my experience, modern models do fairly well for TDDFT calculations. Be careful about the atomic radii used and whether the computational program generates a reasonable solvation surface.

Also keep in mind that not every program supports both D-PCM and C-PCM. Many packages just implement one solvation model.

There are a number of excellent reviews on polarizable continuum solvation models. As you're probably aware, they are a really efficient way to handle molecular solvation in calculations, since the details of the solvent are largely ignored. Instead, the solvent is treated as a dielectric environment, and you handle polarization between the molecule and the solvent self-consistently. That is, the solute molecule is treated at normal atomistic detail, creates a solvation shell (usually treated by a set of overlapping spheres) and the electrostatics of the molecule influences the solvent, and vice-versa.

Now, I've never seen "C-PCM" as a "capacitive" PCM. Usually people refer to the "C" as a conductor-style model. That is, in a C-PCM model, the dielectric constant is set to $\infty$ but then you re-scale the charges on the molecule later.

There are more than just these techniques. The main problem is that it's very hard to bench-mark solvation energies.

Some great recent reviews:

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