In this paper, alpha power transformation for continuous distributions is adapted to discrete distributions. The new family for discrete distributions called discrete alpha power transformation is proposed. The discrete alpha power transformation-exponential distribution is studied in detail. Several distributional properties of introduced distribution including moments, survival and hazard rate functions, mode, and quantile function are discussed. The statistical inference on the model parameters is studied by maximum likelihood, moments, and least-squares estimation methods. A simulation study is performed to observe the performance of bias and mean square errors of these estimates. Three bootstrap methods are considered for constructing confidence intervals for the distribution parameters. As an application of the discrete alpha power transformation-exponential distribution, a new zero-inflated count regression model is proposed to be an alternative model for zero-inflated Poisson, zero-inflated geometric, and zero-inflated negative binomial regression models. Two examples with real data are provided to illustrate the applicability of introduced distribution and count regression analysis.