Fast radio bursts are millisecond-duration astronomical radio pulses of unknown physical origin that appear to come from extragalactic distances. Previous follow-up observations have failed to find additional bursts at the same dispersion measure (that is, the integrated column density of free electrons between source and telescope) and sky position as the original detections. The apparent non-repeating nature of these bursts has led to the suggestion that they originate in cataclysmic events. Here we report observations of ten additional bursts from the direction of the fast radio burst FRB 121102. These bursts have dispersion measures and sky positions consistent with the original burst. This unambiguously identifies FRB 121102 as repeating and demonstrates that its source survives the energetic events that cause the bursts. Additionally, the bursts from FRB 121102 show a wide range of spectral shapes that appear to be predominantly intrinsic to the source and which vary on timescales of minutes or less. Although there may be multiple physical origins for the population of fast radio bursts, these repeat bursts with high dispersion measure and variable spectra specifically seen from the direction of FRB 121102 support an origin in a young, highly magnetized, extragalactic neutron star.