Sharp, nonasymptotic bounds are derived for the best achievable error probability in binary hypothesis testing between two probability distributions with independent and identically distributed observations. The asymmetric version of the problem is considered, where different requirements are placed on the two error probabilities. Using techniques from large deviations theory and normal approximation, accurate nonasymptotic expansions are obtained with explicit constants. Examples are shown indicating that, in the asymmetric regime, the approximations suggested by the new bounds are significantly more accurate than the approximations provided by either of the two main earlier approaches – normal approximation and error exponents.