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We study the importance of baryonic physics on predictions of the matter power spectrum as it is relevant for forthcoming weak lensing surveys. We quantify the impact of baryonic physics using a set of three cosmological numerical simulations. Each simulation has the same initial density field, but models a different set of physical processes. The first simulation evolves the density field using gravity alone, the second includes non-radiative gasdynamics, and the third includes radiative heating and cooling of baryons, star formation, and supernova feedback. We find that baryonic processes alter predictions for the matter power spectrum significantly relative to models that include only gravitational interactions. Our results imply that future weak lensing experiments such as LSST and SNAP will be very sensitive to the poorly-understood physics governing the nonlinear evolution of the baryonic component of the universe. The net effect is significantly larger in the case of the model with cooling and star formation, in which case our results imply that contemporary surveys such as the CFHT Wide survey may also be sensitive to baryonic processes. In particular, this effect could be important for forecasts of the constraining power of future surveys if information from scales larger than l ~ 1000 is included in the analysis. We find that deviations are caused primarily by the rearrangement of matter within individual dark matter halos relative to the gravity-only case, rather than a large-scale rearrangement of matter. Consequently, we propose a simple model, based on the phenomenological halo model of dark matter clustering, for baryonic effects that can be used to aid in the interpretation of forthcoming weak lensing data.