The polarization of light in Raman spectroscopy is important due to selection rules.

With polarized light your Raman spectra will only have some of Raman active modes. If you rotate the polarization you can gain access to the other modes. The modes will be separated accordingly to their symmetry. For example, if for one polarization you see a mode from the Ag symmetry, you will see every mode from that symmetry with that polarization (see the attachment for an example).

When you use non-polarized light your Raman spectra will be the sum the ones before, you will see every Raman active-modes at the same time. regardless their symmetry. (In the attached example, with unpolarized light you would have the sum of upper and lower spectra).

There is a good intro video about polarization in Raman measurements:

http://www.youtube.com/watch?v=53wsLZ5FgbM

Polarised light is very useful to probe oriented materials

"Polarized Photoluminescence and Measurement of Degree of Order in Mixed Oriented Polymer Liquid Crystal Films", A.P. Davey, R.G. Howard, W.J. Blau and H.J. Byrne, Int. J. of Polymeric Materials, 44, 241 (1999).

Either the sample or the polarisation can be rotated, but be careful in rotating the polarisation as often the gratings are polarisation sensitive.

Symmetry of the vibrational modes are very important in vibrational spectroscopy. In the case of Raman spectroscopy measurements, you can differentiate the symetric modes from the others by ussing polarized light. In this way, information about the symmetries of the sample ca be obtained.

Symmetry band assignment and average molecular orientation can be determined if target molecule is partially oriented in streched polymers or in liquid crystals.

In case of gaseous samples:

The orientatio of the laser polarisation with respect to the scattering plane (defined by laser beam, scattering target and line of sight of collection optics) will determine the intensity of the scattered light as the intensity distribution is not isotropic. So chose the polarisation wisely if you want to maximise the Raman signal...

This effect is directly used in measurements of the depolarisation ratios of molecular species, see e.g.

"Accurate depolarization ratio measurements for all diatomic hydrogen isotopologues", T.M. James, M. Schlösser, S. Fischer, M. Sturm, B. Bornschein, R. J. Lewis, and H. H. Telle, Journal of Raman Spectroscopy, 44 6, 857—865 (2013), doi:10.1002/jrs.4283

All the responses are correct but overlook a couple of simple points. Samples which are crystalline will affect the polarisation of the beam. A simple way to test this is to record a spectrum then turn the sample through 90 degrees. Record the spectrum again, any changes are due to orientation of the crystal. However you should also check with the instrument manufacturer whether the beam is polarised or not. Some use polarised lasers and others use unpolarised lasers

The Polarized Raman and photoluminescence studies of a sub-micron sized hexagonal AlGaN crystallite for structural and optical properties is reported in the following article, it may help you....

https://www.researchgate.net/publication/290123296_Polarized_Raman_and_photoluminescence_studies_of_a_sub-micron_sized_hexagonal_AlGaN_crystallite_for_structural_and_optical_properties

Article Polarized Raman and photoluminescence studies of a sub-micro...

There is one additional complication to the one reported by Geoffrey Dent : If your vibrations are both, Raman and Infrared active, also the scattering geometry (90° vs. 180°) and the orientation of the IR transition moment relative to the incidence plays a role. E.g., if you use 180°, and your are interested in the fully symmetric species, you will notice a blue shift of the bands if incidence direction is parallel to the transition in comparison to the case where the transition moment is perpendicular. The larger the oscillator strength, the stronger is this shift. See e.g.

In a non-polarised Raman measurement how do the crystal orientation affect the signal intensity? Can we relate it to polarisability tensor of each vibrational mode?

For a detailed polarized Raman scattering analysis, which relates laser beam polarization with crystal orientation I like to suggest the following publication, which also answers the recent question by Reshmi Thomas about this topic.

Uma Ramabadran, and Bahram Roughani, “Intensity Analysis of Polarized Raman Spectra for off Axis Single Crystal Silicon”, Mat. Sci. and Eng. B, 230, pp 31–42 (April 2018). https://doi.org/10.1016/j.mseb.2017.12.040

[with a featured review about this paper entitle; “Shining light on the off-axis single crystalline Si properties” that appeared at Advances in Engineering (https://advanceseng.com/)].