Novel vortex structures are found when a thin superconducting (SC) film is covered with a lattice of out-of-plane magnetized magnetic dots (MDs). The stray magnetic field of the dots confines the vortices to the MD regions, surrounded by antivortices which "crystallize" into regular lattices. First- and second-order transitions are found as the magnetic array is made sparser or MD magnetization larger. For sparse MD arrays fractional vortex-antivortex states are formed, where the crystal symmetry is combined with a nonuniform "charge" distribution. We demonstrate that due to the (anti)vortices and the supercurrents induced by the MDs, the critical current of the sample actually increases if exposed to a homogeneous external magnetic field, contrary to conventional SC behavior.