The cerium hexaboride (CeB_{6}) f-electron compound displays a rich array of low-temperature magnetic phenomena, including a "magnetically hidden" order, identified as multipolar in origin via advanced x-ray scattering. From first-principles electronic-structure results, we find that the antiferroquadrupolar (AFQ) ordering in CeB_{6} arises from crystal-field splitting and yields a band structure in agreement with experiments. With interactions of p electrons between Ce and B_{6} being small, the electronic state of CeB_{6} is suitably described as Ce(4f^{1})^{3+}(e^{-})(B_{6})^{2-}. The AFQ state of orbital spins is caused by an exchange interaction induced through spin-orbit interaction, which also splits the J=5/2 state into a Γ_{8} ground state and a Γ_{7} excited state. Within the smallest antiferromagnetic (AFM) (111) configuration, an orbital-ordered AFQ state appears during charge self-consistency, and it supports the appearance of a "hidden" order. Hydrostatic pressure (either applied or chemically induced) stabilizes the AFM (AFQ) states over a ferromagnetic one, as observed at low temperatures.