Using density functional theory (DFT) within the scheme of the linearized augmented plane wave and the improved local orbital (APW+lo), we investigated the newly discovered FeAs layered compounds LaOFeAs, SrFe2As2, CaFe2As2, LiFeAs, NaFeAs and FeSe. We report the crystal structures, magnetism, and electronic properties of these Fe-based materials. The results reveal a close correlation of the superconductivity of these FeAs superconductors with the lattice geometry formation and magnetism of their parent compounds. The maximal critical temperatures Tc could simultaneously increase with the decrease of distortion of the FeAs4 tetrahedrons and the increase of interlayer distance L between two [FeAs] layers. We have further found that the magnetism is mainly attributed to Fe moment, and As 4p and Fe 3d hybrid states affect the physical natures, which are the common feature in these Fe-pnictide superconductors