A good introductory reference text is "Modern Raman Spectroscopy" by E. Smith and G. Dent. This discusses the use of Raman spectroscopy in the laboratory environment and goes over some basic details. A very very short answer to your question is that Raman spectroscopy characterizes a material by showing which vibrational frequencies can be supported by the atomic lattice. For example, the Si-Si bond in pure silicon has a resonant frequency near 520cm-1, which leads to a very sharp feature in the Raman at this frequency. So by shining a laser on the material and observing these scattered frequencies, one can begin to unravel the lattice-scale structure.
In nutshell: If you analyze the intensity of scattered visible light with high precision and high resolution, near to the original frequency (which should be very monochromatic, usually laser light) you will observe maxima at frequencies of original + vibration frequency. Therefore the frequency difference tells you where the vibratonal frequencies are. IR spectroscopy gives similar, but not identical information. The slection rules are different, which is important in highly symmetrical molecules. In IR spectra the dipole moment should change (asymmetric vibrations), while in Raman spectra the polarizability. Another advantage is that Raman can be made in aqueous solutions as well. Disadvantage: fluorescence intrferes with Raman spctroscopy. (That's why usually more laser sources are used). If you use Raman microscopy, you can get higher resolution than in IR microscopy.
In general by using Raman spectroscopy you can measure the Raman signature of the materials. The Raman signature will active if the material has dipole movements. Its works on the basics of light scattering technique .In conventional measurement method sun light were used as a sources for the incident light nowadays laser has made lots of impact on light scattering field so Morden Raman spectroscopy comes with laser as a incidents light source with different wavelength (633nm, 532nm, etc) to acquire the Raman signature for the materials. You must select the appropriate incidents light source to do accurate sample analysis. Silicon and diamond are used as standard materials for Raman spectroscopy. There are lots of book and published papers available in Internet and libraries you can go through it’s a one of the interesting field to works on.
Just a slight revision to Mani's earlier answer: materials are Raman active if the atomic vibration changes the polarizability of the bond. It is FTIR active if the atomic vibration changes or induces a dipole moment in the bond.
Read some text books on Raman spectroscopy first then read some more specialized books like, Light scattering in Solids, ed. M. Cardona and G. Guentherrodt. This is a book series and you will find of all aspect of Raman scattering discussed and explained by specialists. If you interested in molecules only (in gas or liquid state0 then read the classic book of Wilson et al. (Dover publication). All physical chemistry text books also discusses IR absorption and Raman spectroscopy in quite some detail.
The exact reference of the book which gives a very good account of Infrared and Raman spectra is
Molecular Vibrations: Theory of Infrared and Raman Vibrational Spectra, E.B. Wilson, Jr., J.C. Decius and P.C. Cross, Dover Publications, New York (1980).
If you make IR and Raman technique your main tool of research then must read this book throughly before going to other advanced articles.