Understanding the first star formation is critical to cosmology. Matt Turk is a (remarkably young) expert on this. The way they go about it is with computer simulations and checking the results against whatever can be observed. Here is a talk by Matt Turk http://online.kitp.ucsb.edu/online/clusters09/turk/ There are Population III.1 and III.2 All the Pop III stars have primordial composition (mostly 75 % hydrogen and rest helium with trace lithium) in other words no "metals" (heavier atomic species.) The Pop III.1 stars are the ones that formed spontaneously by collapse of dark matter which gradually drew in baryonic matter----and excess energy being radiated by molecular hydrogen. These may be multiple, like binary pair systems. The individual systems tend to be isolated, only one forms in each overdense dark matter region---one (possibly multiple) star per cloud. For clarification see where he discusses this in his talk. The more numerous Pop III.2 stars are ones whose formation has been influenced by earlier stars. For example shockwaves from explosions. Their composition is still primordial but their statistics are different because more different processes contribute to their formation. That's about the first 3% of the talk, then he goes into the simulations of Pop III.1 formation.