We study the polarization anisotropy in the cosmic microwave background (CMB) resulting from patchy reionization of the intergalactic medium by stars in galaxies. It is well known that the polarization of the CMB is very sensitive to the details of reionization, including the reionization epoch and the density fluctuations in the ionized gas. We calculate the effects of reionization by combining a semianalytic model of galaxy formation, which predicts the redshifts and luminosities of the ionizing sources, with a high-resolution N-body simulation, to predict the spatial distribution of the ionized gas. The models predict reionization at redshifts z ~ 5-10, with electron scattering optical depths due to reionization of ~0.014-0.05. We find that reionization generates a peak in the polarization spectrum with amplitude ~0.05-0.15 μK at large angular scales (l ~ 3). The position of this peak reveals the size of the horizon at reionization, whilst its amplitude is a measure of the optical depth to reionization. On small scales (l gtrsim 6000), reionization produces a second-order polarization signal due to the coupling of fluctuations in the free electron density with the quadrupole moment of the temperature anisotropy. Careful study reveals that this coupling generates equal second-order polarization power spectra for the electric and magnetic modes, with amplitude ~10 nK. This amplitude depends strongly on the total baryon density Ωb and on the spatial correlations of the free electron density, and weakly on the fraction fesc of ionizing photons able to escape their source galaxy. The first- and second-order signals are therefore sensitive to different details of how the reionization occurred. Detection of these signals will place important constraints on the reionization history of the universe.