How accurately can 21 cm tomography constrain cosmology?

There is growing interest in using 3-dimensional neutral hydrogen mapping with the redshifted 21 cm line as a cosmological probe. However, its utility depends on many assumptions. To aid experimental planning and design, we quantify how the precision with which cosmological parameters can be measured depends on a broad range of assumptions, focusing on the 21 cm signal from 6 < z < 20. We cover assumptions related to modeling of the ionization power spectrum, to the experimental specifications like array layout and detector noise, to uncertainties in the reionization history, and to the level of contamination from astrophysical foregrounds. We derive simple analytic estimates for how various assumptions affect an experiment's sensitivity, and we find that the modeling of reionization is the most important, followed by the array layout. We present an accurate yet robust method for measuring cosmological parameters that exploits the fact that the ionization power spectra are rather smooth functions that can be accurately fit by 7 phenomenological parameters. We find that for future experiments, marginalizing over these nuisance parameters may provide constraints almost as tight on the cosmology as if 21 cm tomography measured the matter power spectrum directly. A future square kilometer array optimized for 21 cm tomography could improve the sensitivity to spatial curvature and neutrino masses by up to 2 orders of magnitude, to Delta Omega(k)approximate to 0.0002 and Delta m(nu)approximate to 0.007 eV, and give a 4 sigma detection of the spectral index running predicted by the simplest inflation models.