Why is an opaque body opaque? When does a body qualify to be called an opaque body? Is it anybody which cannot let visible light through it or is there any other definition? And when and how does a body allow radiations through it?
 A: There are several terms with precise meaning in physical sciences that have been pulled into common language and misused.
Opaque however does not appear to be one of them. According to several etymology sources I just looked up (like Etymology Online for example) it appears to have rather mushy origins in Latin and French meaning darkened or shady.
For the accepted science definition, I like the American Heritage Science Dictionary's definition
Essentially, it says an opaque object resists transmitting (allowing to pass through) radiation (meaning radiated energy) of some specified sort. So in the example of glass from Guo Qianyi in comments, glass is roughly transparent to visible light but roughly opaque to infrared and ultraviolet light.
A: There are two questions here. The first - "what is the definition of opaque" is terribly broad and depends on the field / context. I will focus on the second: when and how does a body let radiation through?
We should really ask the converse question: by what mechanisms does a body stop radiation from going through. I will answer this for different parts of electromagnetic radiation only.
In the broadest sense, the answer is "when the body interacts with the radiation, some of it will not continue to propagate in the original direction". We differentiate between coherent scatter: radiation changes direction but maintains energy, incoherent scatter: radiation changes direction and loses some energy, and absorption: energy from radiation is transferred entirely to the body. When you have radiation that can conveniently be thought of as particles (that is, wavelength is short compared to the dimension of the body) we usually call them photons (or gamma rays, depending on how they were generated); when the wavelength is much longer, it's more convenient to think of the radiation as a continuous wave.
So let's take them in turn. Photons interact with the electrons in the body. At high energies, the photon energy is much greater than any chemical binding energy of the electrons, and the interaction is virtually indistinguishable from "free electron" interaction. You have Compton scatter (incoherent - photon bounces of electron and transfers some energy / momentum), and sometimes (at high enough energy, i.e. > 1.022 MeV) pair production where a positron/electron pair is created "out of vacuum", robbing the photon of energy in the process. At slightly lower energies you have photoelectric interactions: the bound electron absorbs the energy of the incident photon and jumps to another state. Each of these rob the photon of (some of) its energy. By contrast coherent scatter just changes the direction of the photon. That in itself does not prevent radiation from being transmitted, but as the path length of the radiation increases (it keeps changing direction) any absorption mechanisms have a greater probability of acting on the photon before it gets out.
As you get closer to the visible spectrum, the binding energy of electrons between atoms starts to come into play, and leads to specific absorption peaks - this is what gives color to many molecules. Towards the infrared, energy related to atomic motion (vibrations of bonds, and rotations) become important and lead to more mechanisms for absorbing the energy - this is the cornerstone of IR spectroscopy, of course.
Two further mechanisms worth mentioning: refractive index mismatch, and conductivity. When you have a polycrystalline material, or another material with heterogeneous optical properties (grain boundaries, or lots of material-air interfaces in the case of sintered materials or even loose grained material: think a bowl of sugar) then light has a probability of reflection or refraction at each of these boundaries. This is an example of coherent scatter - no energy lost due to the refraction, in principle - and can lead to a white appearance of the material (again, think bowl of sugar). On the other hand, absorption mechanisms may be in play at the same time (brown sugar), so the two often appear together.
Finally conductivity plays a big role especially at longer wavelengths (lower frequencies); although really all electromagnetic interactions ultimately involve the acceleration of electrons, for long wave lengths you really need macroscopic conductivity (rather than movement of an individual electron) in order to affect the incoming radiation. When conductivity is very good, the electrons will move in such a way as to reflect the radiation; but when conductivity is not quite so good, losses will occur (ultimately resulting in heating of the material). This is almost always a function of frequency of the radiation.
I hope this gave you a bit of a taste - this is really a very broad question. Google some of the terms used above - you can easily spend an afternoon reading about the subject without running the risk of becoming an expert...
A: Optically speaking, and very simply:
An opaque material permits no light to pass through it.
A material which passes light but does not pass image detail is called translucent.
A material which passes light AND image detail is called transparent.
"Passing light" technically means ANY light, but in practice some materials pass so little light that they may effectively be considered opaque, so the term can be subjective in some cases.
A: Opacity is a description of the scattering of light in a certain material: the more opaque a material is, the higher the scattering of light and conversely, the less opaque a material is, the lower the scattering of light.
With a higher scattering of light, the probabilities that light is transmitted through the material diminishes. 
For a more intuitive approach I recommend reading Richard Feynman's QED: The Strange Theory of Light and Matter. 
I use Wikipedia's definition  of scattering where transmission means minimal scattering and where reflection is a form of scattering.
