We introduce a parameterized high-density equation of state (EOS) in order to systematize the study of constraints placed by astrophysical observations on the nature of neutron-star matter. To obtain useful constraints, the number of parameters should be smaller than the number of neutron-star properties that have been measured or will have been measured in the next several years. And the set must be large enough to accurately approximate the large set of candidate EOSs. We find that a parameterized EOS based on piecewise polytropes with 3 free parameters matches to about 4% rms error an extensive set of candidate EOSs at densities below the central density of 1.4 solar mass stars. Adding observations of more massive stars constrains the higher density part of the EOS and requires an additional parameter. We obtain constraints on the allowed parameter space set by causality and by present and near-future astronomical observations. In particular, we emphasize potentially stringent constraints on the EOS parameter space associated with two measured properties of a single star; and we find that a measurement of the moment of inertia of PSR J0737-3039A can strongly constrain the maximum neutron-star mass. We also present in an appendix a more efficient algorithm than has previously been used for finding points of marginal stability and the maximum angular velocity of stable stars.