Laboratory experiments searching for galactic dark matter particles scattering off nuclei have so far not been able to establish a discovery. We use data from the XENON100 experiment to search for dark matter interacting with electrons. With no evidence for a signal above the low background of our experiment, we exclude a variety of representative dark matter models that would induce electronic recoils. For axial-vector couplings to electrons, we exclude cross sections above 6 x 10(-35) cm(2) for particle masses of m(chi) = 2 GeV/c(2). Independent of the dark matter halo, we exclude leptophilic models as an explanation for the long-standing DAMA/LIBRA signal, such as couplings to electrons through axial-vector interactions at a 4.4 sigma confidence level, mirror dark matter at 3.6 sigma, and luminous dark matter at 4.6 sigma.