Using dark matter halos traced by galaxy groups selected from the Sloan Digital Sky Survey Data Release 4, we find that about 1/4 of the faint galaxies (0.1 Mr – 5log h> – 17.05, hereafter dwarfs) that are the central galaxies in their own halo are not blue and star forming, as expected in standard models of galaxy formation, but are red. In contrast, this fraction is about 1/2 for dwarf satellite galaxies. Many red dwarf galaxies are physically associated with more massive halos. In total, about ~45% of red dwarf galaxies reside in massive halos as satellites, while another ~25% have a spatial distribution that is much more concentrated toward their nearest massive halos than other dwarf galaxies. We use mock catalogs to show that the reddest population of nonsatellite dwarf galaxies is distributed within about three times the virial radii of their nearest massive halos. We suggest that this population of dwarf galaxies is hosted by low-mass halos that have passed through their massive neighbors, and that the same environmental effects that cause satellite galaxies to become red are also responsible for the red colors of this population of galaxies. We do not find any significant radial dependence of the population of dwarf galaxies with the highest concentrations, suggesting that the mechanisms operating on these galaxies affect color more than structure. However, over 30% of dwarf galaxies are red and isolated and their origin remains unknown.