Quantum-photonic chips, which integrate quantum light sources alongside active and passive optical elements, as well as single-photon detectors, show great potential for photonic quantum information processing and quantum technology. Mature semiconductor nanofabrication processes allow for scaling such photonic integrated circuits to on-chip networks of increasing complexity. Second-order nonlinear materials are the method of choice for generating photonic quantum states in the overwhelming majority of linear optic experiments using bulk components, but integration with waveguide circuitry on a nanophotonic chip proved to be challenging. Here, we demonstrate such an on-chip parametric down-conversion source of photon pairs based on second-order nonlinearity in an aluminum-nitride microring resonator. We show the potential of our source for quantum information processing by measuring the high visibility anti-bunching of heralded single photons with nearly ideal state purity. Our down-conversion source yields measured coincidence rates of 80 Hz, which implies MHz generation rates of correlated photon pairs. Low noise performance is demonstrated by measuring high coincidence-to-accidental ratios. The generated photon pairs are spectrally far separated from the pump field, providing great potential for realizing sufficient on-chip filtering and monolithic integration of quantum light sources, waveguide circuits and single-photon detectors.