Neutrinos are copiously produced at particle colliders, but no collider neutrino has ever been detected. Colliders produce both neutrinos and anti-neutrinos of all flavors at very high energies, and they are therefore highly complementary to those from other sources. FASER, the Forward Search Experiment at the LHC, is ideally located to provide the first detection and study of collider neutrinos. We investigate the prospects for neutrino studies with FASER Formula omitted, a proposed component of FASER, consisting of emulsion films interleaved with tungsten plates with a total target mass of 1.2 t, to be placed on-axis at the front of FASER. We estimate the neutrino fluxes and interaction rates, describe the FASER Formula omitted detector, and analyze the characteristics of the signals and primary backgrounds. For an integrated luminosity of Formula omitted to be collected during Run 3 of the 14 TeV LHC in 2021-23, approximately 1300 electron neutrinos, 20,000 muon neutrinos, and 20 tau neutrinos will interact in FASER Formula omitted, with mean energies of 600 GeV to 1 TeV. With such rates and energies, FASER will measure neutrino cross sections at energies where they are currently unconstrained, will bound models of forward particle production, and could open a new window on physics beyond the standard model.