The recent detection of variable infrared emission from Sagittarius A*, combined with its previously observed flare activity in X-rays, provides compelling evidence that at least a portion of this object's emission is produced by nonthermal electrons. The polarization and variability of Sgr A*'s emission depend strongly on the observed wavelength, indicating distinct physical processes. We show here that acceleration of electrons by plasma wave turbulence in hot gases near the black hole's event horizon, though with several theoretical uncertainties, can reasonably account both for Sgr A*'s mm and shorter wavelength emission in the quiescent state, and for the infrared and X-ray flares, induced either via an enhancement of the mass accretion rate onto the black hole or by reorganization of magnetic fields coupled to the accretion gas. High-energy electrons diffusing away from the acceleration site toward larger radii might account for Sgr A*'s emission at longer wavelengths. The acceleration model produces prominent IR flares accompanying X-ray bursts. Future coordinated multi-wavelength observations will be able to test this model and constrain its parameters.