The existence of bound states of elementary spin waves (magnons) in one-dimensional quantum magnets was predicted almost 80 years ago(1). Identifying signatures of magnon bound states has so far remained the subject of intense theoretical research(2-5), and their detection has proved challenging for experiments. Ultracold atoms offer an ideal setting in which to find such bound states by tracking the spin dynamics with single-spin and single-site resolution(6,7) following a local excitation(8). Here we use in situ correlation measurements to observe two-magnon bound states directly in a one-dimensional Heisenberg spin chain comprising ultracold bosonic atoms in an optical lattice. We observe the quantum dynamics of free and bound magnon states through time-resolved measurements of two spin impurities. The increased effective mass of the compound magnon state results in slower spin dynamics as compared to single-magnon excitations. We also determine the decay time of bound magnons, which is probably limited by scattering on thermal fluctuations in the system. Our results provide a new way of studying fundamental properties of quantum magnets and, more generally, properties of interacting impurities in quantum many-body systems.