High-energy cosmic neutrinos carry unique information about the most energetic non-thermal sources in the Universe. This white paper describes the outstanding astrophysics questions that neutrino astronomy can address in the coming decade. A companion white paper discusses how the observation of cosmic neutrinos can address open questions in fundamental physics. Detailed measurements of the diffuse neutrino flux, measurements of neutrinos from point sources, and multi-messenger observations with neutrinos will enable the discovery and characterization of the most energetic sources in the Universe.