Recent infrared (IR) observations of the center of our Galaxy indicate that the supermassive black hole source Sgr A* is strongly variable in the IR. The timescale for the variability, 30 min, is comparable to that of the X-ray flares observed by Chandra and XMM, suggesting a common physical origin. In this paper, we investigate the nature of the IR emission in the context of models recently proposed to interpret the X-ray flares. We show that the IR emission in Sgr A* can be well explained by nonthermal synchrotron emission if a small fraction of the electrons in the innermost region of the accretion flow around the black hole are accelerated into a broken power-law distribution, perhaps due to transient events such as magnetic reconnection. The model predicts differences in the variability amplitudes of flares in the IR and X-rays, in general agreement with observations. It also predicts that the IR emission should be linearly polarized, as has indeed been observed during one epoch. IR and X-ray flares analogous to those observed in Sgr A* may be detectable from other accreting SMBHs, provided L < 10-8 L_EDD; at higher luminosities the flaring emission is dominated by thermal synchrotron-self Compton emission, which is likely to be less variable.