High-strength metal alloys achieve their performance via careful control of precipitates and solutes. The nucleation, growth, and kinetics of precipitation, and the resulting mechanical properties, are inherently atomic-scale phenomena, particularly during early-stage nucleation and growth. Atomistic modeling using interatomic potentials is a desirable tool for understanding the detailed phenomena involved in precipitation and strengthening, which requires length and time scales far larger than those accessible by first-principles methods. Current interatomic potentials for alloys are not, however, sufficiently accurate for such studies. Here, a family of neural-network potentials (NNPs) for the Al-Cu system is presented as a first example of a machine-learning potential that can achieve near-first-principles accuracy for many different metallurgically-important aspects of this alloy. High fidelity predictions of intermetallic compounds, elastic constants, dilute solid-solution energetics, precipitate/matrix interfaces, generalized stacking fault energies, and surfaces for slip in matrix and precipitates, antisite defect energies, and other quantities, are shown. The NNPs also capture the subtle entropically-induced transition between θ' and θ at temperatures around 600K. Many comparisons are made with the state-of-the-art Angular-Dependent Potential for Al-Cu, demonstrating the significant quantitative benefit of a machine-learning approach. A preliminary kinetic Monte Carlo study shows the NNP to predict the emergence of GP zones in Al 4%Cu at T=300K in agreement with experiments. The NNP shows some significant transferability to defects and properties outside the structures used to develop the NNP but also shows some errors, highlighting that the use of any interatomic potential requires careful validation in application to specific metallurgical problems of interest.