All the high-temperature superconductors have copper oxide planes with strong antiferromagnetic correlations. In many of these, the charge is carried by quasiparticles which are based on holes that move on the oxygen ions. We study the effect of the antiferromagnetic background of the Cu spins in the CuO2 planes on the spectrum of the mobile quasiparticles. We consider several possible quasiparticles, which involve the spin of the hole and the spins of the two neighboring copper ions. For each quasiparticle we obtain, using tight-binding methods, four energy bands. We find that the favored charge carrier is a spin 1/2 quasiparticle. The minimum of the lowest energy band is at the points k=(0,/2a) and k=(/2a,0), where the x and y directions are along the nearest-neighbors oxygen-oxygen bonds (of length a) in the CuO2 plane. The effective mass is less than 3.7me, in good agreement with measurements of the London penetration depth for La2-xSrxCuO4. It is also consistent with these measurements for the Bi and Tl compounds. However, we obtain conflicting results for YBa2Cu3O6+y, possibly due to the fact that the CuO chains also superconduct.