Centrosymmetric is related to the site symmetry in which a certain atom is placed. Such site symmetry contains an inversion center as one of its symmetry elements. The inversion center can hinder some properties in the crystals.

There is different ways to identify which material is a Centrosymmetric or non-Centrosymmetric system. The average structure can be checked by X-ray powder diffraction that will provide the point group of the crystal system. For a more precise information one can use EXAFS analyses in order to check eventual disorders in your point symmetry. Other efficient way to determine the site symmetry of a determined system is to study the photoluminescent spectroscopy of Eu3+ doped materials. The Eu3+ is a spectroscopic probe for Centrosymmetric environment having the electric dipole transitions supressed in such symmetry.

An additional option could be vibrational spectroscopy. For centrosymmetric crystals and molecules IR-active vibrations are not Raman-active and vice versa (mutual exclusion principle). In contrast, if no inversion center is present, IR-active vibrations are also Raman-active (but not all Raman-active vibrations are also IR-active!).

Dear Pria there is beautiful note on the subject in WIKIPEDIA  under the title of 'Centrosymmetric'

Actually it is a point group in crystallography which has an inversion center as one of its symmetry element. That means any any atomic species is represented by   R   vector in the unit cell there  correspond identical  atomic species station in that  point  denoted by  -R vector.

Point group lacking the inversion symmetry is called non-centrosymmetric_ . Which   are further divided into polar and chiral types.  See please above  cited notes to get more details.

Under laser irradiation the noncentrosymmetric crystal does generate second harmonic, and the centocymmetric does not. 

http://pubs.acs.org/doi/abs/10.1021/ja9061568