Graphene near charge neutrality is expected to behave like a quantum-critical, relativistic plasma—the "Dirac fluid"—in which massless electrons and holes rapidly collide at a rapid rate. We measure the frequency-dependent optical conductivity of clean, micron-scale graphene at electron temperatures between 77 and 300 K using on-chip terahertz spectroscopy. At charge neutrality, we observe the quantum-critical scattering rate characteristic of the Dirac fluid. At higher doping, we uncover two distinct current-carrying modes with zero and nonzero total momenta, a manifestation of relativistic hydrodynamics. Our work reveals the quantum criticality and unusual dynamic excitations near charge neutrality in graphene.