the Beer-Lambert law is what you are looking for: absorbance is proportional to concentration.

care must be taken when analyzing a liquid system in which strong molecular intractions are going on. such interactions may alter the dipol moments and hence result in intensity nonlinearities and frequency shifts of the FTIR peak

You have to use the Lambert-Beer law as Johannes Kiefer wrote. The second problem is that you have to keep the absorbance in the region of LB law's linearity. This means that the absorbance should be about 1 max 2 in the transmission measurement and reflectance lower then 1 in ATR.

Beer-Lambert law requires information about the depth penetration, refractive indexes..etc 

Here, my aim is to get use of the area under the peak by applying integration for the target peak. I have heard before that this area is related somehow to the concentration

One more thing: my experiment is insituFTIR, which means that the analyzed peak is a result of  the interaction between the adsorbent and the adsorbate (physical adsorption) which, in principle, should undergo a change in position and/or shape based on that interaction

the most straightforward method would be to carry out a calibration measurement and plot the peak area (or max absorbance of peak) as a function of concentration. such a calibration usually even works for overlapping/shifting peaks.

I assume your insituFTIR is an ATR probe. ATR spectra are not easy to quantify without some calibration. The parameter you would need to determine with reasonably good accuracy is not the penetration depth but the effective path length, see e.g. DOI: 10.1021/ac7025892 or http://dx.doi.org/10.1016/j.memsci.2013.10.022

In literature, ATR is the most widely used method in this regard. However, I use DRIFTS (Diffuse Reflectance Infrared Fourier Transform Spectroscopy); so how this would affect the way I analyze the concentration?

what kind of sample do you look at? if the surface is rough and not homogenoeus, the quantification of DRIFTS spectra is quite painful

As you say, quantification in spectroscopy is based on peak heights, not areas. To use the area instead, you must consider what factors could be determining the observed width of the band. Instrument resolution? Overlap with other bands? Rotational substructure? Only if you can exclude any of that is it legitimate to use areas instead of peak amplitudes. Simple test: plot the spectrum at various concentrations, scaled to the same peak height. Is the shape of the band then identical? If not, you can't use the area for quantification.

Also, as Johannes says, reflection from a surface is not the method of choice for quantitative spectroscopy. Assuming you're trying to quantify some material in a surface layer, the problem is that increasing the "concentration" of that material will also increase the layer thickness. There must come a point where the layer is too thick to be penetrated fully by the IR beam in a reflection measurement: from then on, you can add further material without changing the spectrum, so the method is non-quantitative by definition. You also have to consider the changing contribution of the underlying substrate to the spectrum as the layer thickness increases.

I agree with Johannes, 

I agree with Moss. S/N should be well for the quantification purpose. If you see overlap, peak or curve fitting is also important. DRIFTS spectra are always rough; increasing scanning # is another solution.

Under in-situ FTIR, peak shape/position are shifting. Did you identify a peak that is relatively stable for both its shape (intensity) and position of course, and then you can use this peak as a reference peak. Taking ratios of areas/height of your interested peak to that of the reference peak during quantification. Again, if your peak shifted, you may want to do peak resolve. 

Good luck.

I think this link is very useful, 

http://www.instanano.com/characterization/theoretical/concentration-calculation-from-uv-vis-absorbance/

And you should not forget that the absorbance you get from -lg T or -lg(R/R0) yet needs to be corrected before you can apply Beer's law, see e.g. Article Employing Theories Far beyond Their Limits-The Case of the (...

Article The Electric Field Standing Wave Effect in Infrared Transmis...

Article The Electric Field Standing Wave Effect in Infrared Transmis...

Article The Electric Field Standing Wave Effect in Infrared Transmis...

In the meantime there has been established a connection between electromagnetic theory and Beer's law/approximation. In fact, Beer's approximation can be derived from the Lorentz-Lorenz theory under certain approximations. One is that matter does not polarize under the influence of light, so that the local electric field at the location of a molecule is the same as the one externally applied. Under these circumstances it was shown that the integral of an absorbance band is proportional to the concentration, but not the peak value. For Beer's approximation to hold, you have to assume that x2 is approximately 2x-1 where x is the electric susceptibility. Read the whole story in this recent review (and the papers therein):

Article The Bouguer-Beer-Lambert Law: Shining Light on the Obscure

Preprint Wave optics in Infrared Spectroscopy

If you are interested in the details, see: Article Beer’s law – why integrated absorbance depends linearly on c...