Understanding the structure of quantum correlations in a many-body system is key to its computational treatment. For fermionic systems, correlations can be defined as deviations from Slater determinant states. The link between fermionic correlations and efficient computational physics methods is actively studied but remains ambiguous. We make progress in establishing this connection mathematically. In particular, we find a rigorous classification of states relative to k-fermion correlations, which admits a computational physics interpretation. Correlations are captured by a measure ω_k, a function of k-fermion reduced density matrix that we call twisted purity. A condition ω_k = 0 for a given k puts the state in a class G_k of correlated states. Sets G_k are nested in k, and Slater determinants correspond to k = 1. Classes G_k=O(1) are shown to be physically relevant, as ω_k vanishes or nearly vanishes for truncated configuration-interaction states, perturbation series around Slater determinants, and some nonperturbative eigenstates of the 1D Hubbard model. For each k = O(1), we give an explicit ansatz with a polynomial number of parameters that covers all states in G_k. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.