The dynamic relationship between prey and predator has long been and will continue to be a dominant theme in ecology because of its universality. The prey-predator interaction, one of the most fundamental interspecies interactions, was first described mathematically by Lotka and Volterra in two independent works, resulting in what are now called the Lotka-Volterra equations. A predator-prey model based on the logistic equation was initially proposed by Alfred J. Lotka in 1910 to describe autocatalytic reactions. He later developed this model and in 1925 arrived at the Lotka-Volterra equations that we know today. Almost at the same time (1926), Vito Volterra, an Italian mathematician, independently established the Lotka-Volterra model after analyzing statistical data of fish catches in the Adriatic. The Lotka-Volterra equation is one of the fundamental population models in theoretical biology. Since these early works, prey-predator interactions have been studied systematically. Much of this work has focused on models with continuous time delay as well as their stability, oscillations, Hopf bifurcations and limit cycles, but no attention has been paid to models with piecewise constant arguments and a time delay. In fact, because of environmental factors or predator characteristics, prey are often captured only during certain times of the season. In addition, there is a time delay before hunting because of predator maturation times in practical predator-prey systems. Therefore, it is more realistic to employ the functional response with piecewise constant arguments and a time delay in predator-prey models. In this paper, we discuss the stability and bifurcations of predator-prey systems with piecewise constant arguments and a time delay. First, a discrete model that can equivalently describe the dynamical behavior of the original differential model is deduced. Sufficient conditions for the local asymptotic stability of the steady state are achieved based on an analysis of the eigenvalues and Schur-Cohn criterion. Second, by choosing a parameter r, the intrinsic growth rate of prey, as the bifurcation parameter and using the bifurcation theory and center manifold, we find that the discrete model undergoes a Neimark-Sacker bifurcation at an exceptive value of r. The results show that 1) the stability of the predator-prey system is very complex when we consider piecewise constant arguments and a time delay; and 2) the positive equilibrium of the model switches from being stable to unstable as the intrinsic growth rate of prey increases beyond a critical value, at which point the unique supercritical Neimark-Sacker bifurcation will occur. Finally, computer simulations based on the system supported our main results and illustrated them intuitively. The numerical examples also justify the reasonableness of the conditions given in our paper for the loss of equilibrium. The parameters of the predator-prey model come from nature. However, we can still add to the model a feedback control factor and interference from outside to change the equilibrium, bifurcation point, or amplitude of the periodic solution. Study of our model and its ameliorated version can provide a theoretical basis for understanding ecology and protecting the environment.