The Feenberg-Jastrow Euler-Lagrange theory for inhomogeneous quantum liquids is developed to study many-electron atomic systems. We describe the ground state of the atoms by the Jastrow-Feenberg ansatz for the wave function. The generalized Born-Green-Yvon and Fermi-hypernetted-chain equations are used to relate the wave function to reduced distribution functions. The variational wave function is optimized by a generalized Hartree-Fock equation for the single-particle orbitals and an Euler-Lagrange equation for the two-body correlations. We calculate the electron correlation energies for various atoms and ions with four and ten electrons. Our theoretical results for the correlation energy of the ten-electron systems are within 93% of the experimental data. We discuss the behavior of the exchange-correlation holes in the atomic systems and analyze the qualitative difference of the correlations between the localized and extended states. The theory does not depend on any input parameters and only involves the Coulomb interaction among electrons and the nuclear charge. The relationship of this theory to density functional theory and configuration interaction theory is discussed.