Let 0 < α < 1. We obtain the necessary and sufficient conditions for the boundedness of discrete fractional Hardy-Littlewood maximal operators M α on discrete weighted Lebesgue spaces. From this and a discrete variant of Whitney decomposition theorem, the necessary and sufficient conditions for the boundedness of the discrete Riesz potentials I α on discrete weighted Lebesgue spaces are discussed. As applications, the boundedness of I α on discrete weighted Morrey spaces is further obtained.