On the basis of a well-established binomial structure and the so-called Poisson-Lindley distribution, a new two-parameter discrete distribution is introduced. Its properties are studied from both the theoretical and practical sides. For the theory, we discuss the moments, survival and hazard rate functions, mode and quantile function. The statistical inference on the model parameters is investigated by the maximum likelihood, moments, proportions, least square, and weighted least square estimations. A simulation study is conducted to observe the performance of the bias and mean square error of the obtained estimates. Then, applications to two practical data sets are given. Finally, we construct a new flexible count data regression model called the binomial-Poisson Lindley regression model with two practical examples in the medical area.