When a low-energy electron is incident on an interface between a metal and superconductor, it causes the injection of a Cooper pair into the superconductor and the generation of a hole that reflects back into the metal-a process known as Andreev reflection. In confined geometries, this process can give rise to discrete Andreev bound states (ABS), which can enable transport of supercurrents through non-superconducting materials and have recently been proposed as a means of realizing solid-state qubits. Here, we report transport measurements of sharp, gate-tunable ABS formed in a superconductor-quantum dot (QD)-normal system realized on an exfoliated graphene sheet. The QD is formed in graphene beneath a superconducting contact as a result of a work-function mismatch. Individual ABS form when the discrete QD levels are proximity-coupled to the superconducting contact. Owing to the low density of states of graphene and the sensitivity of the QD levels to an applied gate voltage, the ABS spectra are narrow and can be continuously tuned down to zero energy by the gate voltage.