The basic ingredients for approximation by a negative binomial distribution using Stein's method and a birth and death characterizing equation are established. These include good bounds for the solution of the Stein equation and for its first differences, and a coupling representation of the accuracy of approximation. Lower bounds for the approximation error are also given for cases where a monotone coupling exists. The method is applied to the approximation of the Pólya distribution by a negative binomial.