Population III (Pop III) stars, defined by their extremely low metallicities, are the first sources of light in the universe after the big bang. They begin the processes of metal enrichment and reionization in the universe. For my thesis, I have studied various processes that delay their formation. A critical dark matter halo mass for Pop III stars can be defined as the typical minimum halo mass that hosts sufficient cold dense gas required for the formation of the first stars. We performed a comprehensive cosmological simulation suite to understand the dependence of soft UV background, streaming velocity between baryons and dark matter, and redshift on the critical mass. We provide a fit for this dependence which can be used in semi-analytic models of early galaxy formation. In halos exposed to strong ionizing radiation from nearby galaxies, star formation can be suppressed until halo becomes much more massive. We studied fragmentation in these low redshift massive Pop III galaxies to estimate their total stellar content. Pop III star formation also depends on the nature of dark matter. In particular, I will present our preliminary results on the formation of Pop III stars in the fuzzy dark matter cosmology where dark matter is made up of ultra-light axions and has a small-scale cutoff in the power spectrum.