Particle-vortex duality is a powerful theoretical tool that has been used to study bosonic systems. Here, we propose an analogous duality for Dirac fermions in 2+1 dimensions. The physics of a single Dirac cone is proposed to be described by a dual theory, QED3, with again a single Dirac fermion but coupled to a gauge field. This duality is established by considering two alternate descriptions of the three-dimensional topological insulator (TI) surface. The first description is the usual Dirac fermion surface state. The dual description is accessed via an electric-magnetic duality of the bulk TI coupled to a gauge field, which maps it to a gauged chiral topological insulator. This alternate description ultimately leads to a new surface theory, QED3, which provides a simple description of otherwise intractable interacting electronic states. For example, an explicit derivation of the T-Pfaffian state, a proposed surface topological order of the TI, is obtained by simply pair condensing the dual fermions. The roles of time-reversal and particle-hole symmetries are exchanged by the duality, which connects some of our results to a recent conjecture by Son on particle-hole symmetric quantum Hall states.