The functional and mechanical properties of a material are often determined by the defects in them. A thorough understanding of the relationship between the defects and the properties allows for tailoring a material’s properties into the desired combinations. Amongst the different classes of defects, experimental identification of point defects is typically difficult and indirect, usually requiring an ingenious combination of different techniques. In this context, first-principles calculations, complemented with experiments, offer insights into the formation of defects and their role in properties. This was demonstrated in this thesis through investigations on the effect of calcium vacancies on structure, vibrational and elastic properties hydroxyapatite (HAp), and oxygen vacancies on elastic properties of zinc oxide (ZnO) using first-principles calculations based on density functional theory (DFT). Our results confirm a considerable reduction in the elastic constants of HAp—the inorganic constituent of bone—due to Ca-deficiency, which was experimentally reported earlier. Elastic anisotropic behavior of stoichiometric and Ca-deficient HAp is analyzed, which will be useful in understanding the effects of crystal orientation in designing synthetic bone. Local structural stability of HAp and Ca-deficient HAp structures is assessed with full phonon dispersion studies and the specific signatures in the computed vibrational spectra for Ca deficiency in HAp can be utilized in experimental characterization of different types of defected HAp. In ZnO, formation energies of oxygen vacancies in different types of oxygen deficient structures are analyzed to ascertain their stability. Our results show considerable degradation of some of the elastic moduli due to the presence of such vacancies. Further, the charge state of the defect structure is found to influence the shear elastic constants. Evaluation of elastic anisotropy of stoichiometric and oxygen deficient ZnO indicates the significant anisotropy in elastic properties and stiff c-axis orientation. The second part of the thesis deals with developing some understanding of the pressure-induced phase transformations (PIPT) in an inorganic material, titanium nitride (TiN), and in a metal-organic framework (MOF), erbium formate crystal. PIPT, which is a common phenomenon in many materials, is of great interest in materials science as the properties of the transformation product can diverge significantly from those of the parent phase. Hence, it is important to understand the pressure induced changes so to improve the component reliability and to enhance service life of materials used in high pressure applications. TiN undergoes PIPT from NaCl to CsCl structure. On the basis of our DFT calculations, we propose a new transformation path, which shows that the stress required for this transformation is substantially lower when it is deviatoric in nature than that under hydrostatic pressure. Local stability of the structure is assessed with phonon dispersion determined at different pressures, and we find that CsCl structure of TiN is expected to distort after the transformation. Further, we provide a quantitative comparison of electronic structure of TiN in NaCl structure with that of high pressure phase with implication to electrical conduction properties. Next, we investigate the PIPT associated with bond rearrangement in erbium formate framework. Phase transition pressure is estimated and the corresponding changes in bonding characteristics are analyzed. Estimated lattice constants for both the phases agree well with the earlier experimental results. While the transformation pressure of the framework is overestimated with respect to experiment, our calculations confirm PIPT, and thus provide a theoretical evidence for the experimental finding.