I've seen Raman spectra, such as here, where peaks at certain wavenumbers were identified with phonon vibration modes (TO, LO, TA, LA). What is the relationship behind this, and how could I make such associations? Would the process be different for Raman Spectra involving different types of silicon such as crystalline silicon, amorphous silicon, polycrystalline silicon, etc?
Raman process can be described as inelastic light scattering - that is, part of the incident photon energy goes into internal degrees of freedom of the scattering medium. By looking at the energy spectrum of scattered photons one can deduce energies of these internal degrees - in your case these are phonon vibration modes.
Identifying modes - i.e. saying whether a peak is TO, LO or some other mode - is not straightforward and usually requires prior knowledge (maybe, someone measured it before) or ab initio calculations.