Starting from the characterization of connected, simply connected and complete homogeneous pseudo-Riemannian manifold in terms of a (1,2) tensor field \(S\) on a manifold [\textit{P. M. Gadea} and \textit{J. A. Oubiña}, Houston J. Math. 18, No. 3, 449-465 (1992; Zbl 0760.53029)], the authors obtain a classification of homogeneous pseudo-Riemannian structures into eight classes according to the structure \(S\) belonging to an invariant subspace of a certain space \(S_1 \oplus S_2 \oplus S_3\). After that it is shown that if a connected, simply connected and complete pseudo-Riemannian manifold is a nonflat space form, then it is locally isometric to a manifold which admits a nondegenerate homogeneous structure of class \(S_1\). They also prove that if a connected pseudo-Riemannian manifold of any signature admits a nondegenerate homogeneous structure of class \(S_1\), then it is a nonflat pseudo-Riemannian space form.