We report the first measurements of the moments - mean (M), variance (2), skewness (S), and kurtosis (κ) - of the net-charge multiplicity distributions at midrapidity in Au+Au collisions at seven energies, ranging from sNN=7.7 to 200 GeV, as a part of the Beam Energy Scan program at RHIC. The moments are related to the thermodynamic susceptibilities of net charge, and are sensitive to the location of the QCD critical point. We compare the products of the moments, 2/M, S, and κ2, with the expectations from Poisson and negative binomial distributions (NBDs). The S values deviate from the Poisson baseline and are close to the NBD baseline, while the κ2 values tend to lie between the two. Within the present uncertainties, our data do not show nonmonotonic behavior as a function of collision energy. These measurements provide a valuable tool to extract the freeze-out parameters in heavy-ion collisions by comparing with theoretical models. © 2014 American Physical Society.