  The formation of the first stars marks a watershed moment in the history of
our universe. As the first luminous structures, these stars (also known as
Population III, or Pop III stars) seed the first galaxies and begin the process
of reionization. We construct an analytic model to self-consistently trace the
formation of Pop III stars inside minihalos in the presence of the fluctuating
ultraviolet background, relic dark matter-baryon relative velocities from the
early universe, and an X-ray background, which largely work to suppress cooling
of gas and delay the formation of this first generation of stars. We
demonstrate the utility of this framework in a semi-analytic model for early
star formation that also follows the transition between Pop III and Pop II star
formation inside these halos. Using our new prescription for the criteria
allowing Pop III star formation, we follow a population of dark matter halos
from $z=50$ through $z=6$ and examine the global star formation history,
finding that each process defines its own key epoch: (i) the stream velocity
dominates at the highest redshifts ($z\gtrsim30$), (ii) the UV background sets
the tone at intermediate times ($30\gtrsim z\gtrsim15$), and (iii) X-rays
control the end of Pop III star formation at the latest times ($z\lesssim 15$).
In all of our models, Pop III stars continue to form down to $z\sim 7-10$, when
their supernovae will be potentially observable with forthcoming instruments.
Finally, we identify the signatures of variations in the Pop III physics in the
global 21-cm spin-flip signal of atomic hydrogen.
