A landmark discovery was made in 2003 when the first binary system containing two active radio pulsars was observed. Not only is this system, PSR J0737-3039A/B, the most relativistic binary system ever found, but it also exhibits eclipses and magnetospheric interactions. The observational and theoretical insights provided by this first double pulsar go far beyond applications in pulsar astrophysics alone. Timing observations of the two pulsars provide the best test of general relativity (GR) in strong gravitational fields. Studies of the system's evolution suggest the possibility of an unusual neutron star formation process. The discovery of this system has a significant impact on the expected coalescence rate of galactic double neutron stars and the detection of merging systems in Earth-bound gravitational-wave detectors. Moreover, observations and modeling of the eclipses provide evidence for relativistic spin precession and dipolar magnetic fields in pulsar magnetospheres. This paper reviews the double pulsar's properties and what they mean for fundamental physics and astrophysics.