The first usage is as a synonym for refractive index, as described in the answers to the related question you cite. This is the commoner usage in physics.

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I'm a zoology minor and we are doing protein estimation by colorimetric method. I have stumbled upon a term 'Optical density'. I don't understand the term well. Is it a measure of the extent of light that can pass through a particular object?

For all intents and purposes, OD is the negative of the order of magnitude the factor by which the intensity of the light is reduced by the attenuating element with said OD.

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For example, laser goggles marked $OD7\;488{\rm nm}$ means that the goggles will reduce incident power at 488nm by a factor of $10^7$. Goggles marked with a lone wavelength rather than a wavelength range are always meant for use with a particular kind of laser. For example, the $OD7\;488{\rm nm}$ goggles are meant for use with an argon ion laser. You cannot rely on them using another source of wavelength 485nm, for example.

For example, an optical density of 3 means that the optical power is attenuated by the factor $10^3$ That would correspond to an attenuation by 30 decibels.

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For generic use, a wavelength range must be specified. So, for example, one often sees $OD7\;450{\rm nm} - 510{\rm nm}$, meaning, pretty obviously, goggles that will give you seven orders of magnitude of attenuation over the whole range $450{\rm nm} \leq \lambda \leq 510{\rm nm}$.

To further the explanation above, I thought this image describing the second usage of optical density may be useful. The concept is being applied to a lens in regards to a laser passing through it. Each wavelength in the following chart will have its own OD but the notation is stating the minimum optical density over a range of wavelengths.

In other words: OD = 6, means that the intensity will be reduced by a factor of 10 to the power of -6, a.k.a by a factor of a million.

Optical density may also mean the absolute value of the logarithm with base 10 of the power transmission factor of an optical attenuator:

The second usage is the total attenuation afforded by a protective screen, neutral density filter, laser goggles or the like. $ODx\; \lambda=y$ or even $ODx\; y$ means that the filter, goggles etc afford a power attenuation factor of $10^x$ at a light wavelength of $y$ or light wavelength range $y$. That is, the power transmitted through the filter is $10^{-x}$ of the incident power when the wavelength is as stated.