  We study the potential impact of improved future supernovae data on our
understanding of the dark energy problem. We carefully examine the relative
utility of different fitting functions that can be used to parameterize the
dark energy models, and provide concrete reasons why a particular choice (based
on a parameterization of the equation of state) is better in almost all cases.
We discuss the details of a representative sample of dark energy models and
show how future supernova observations could distinguish among these. As a
specific example, we consider the proposed ``SNAP'' satellite which is planned
to observe around 2000 supernovae. We show how a SNAP-class data set taken
alone would be a powerful discriminator among a family of models that would be
approximated by a constant equation of state for the most recent epoch of
cosmic expansion. We show how this family includes most of the dark energy
models proposed so far. We then show how an independent measurement of
$\Omega_{\rm m}$ can allow SNAP to probe the evolution of the equation of state
as well, allowing further discrimination among a larger class of proposed dark
energy models. We study the impact of the satellite design parameters on this
method to distinguish the models and compare SNAP to alternative measurements.
We establish that if we exploit the full precision of SNAP it provides a very
powerful probe.
