  Quantum nonlocality of several four-qubit states is investigated by
constructing a new Bell inequality. These include the
Greenberger-Zeilinger-Horne (GHZ) state, W state, cluster state, and the state
$|\chi>$ that has been recently proposed in [PRL, {\bf 96}, 060502 (2006)]. The
Bell inequality is optimally violated by $|\chi>$ but not violated by the GHZ
state. The cluster state also violates the Bell inequality though not
optimally. The state $|\chi>$ can thus be discriminated from the cluster state
by using the inequality. Different aspects of four-partite entanglement are
also studied by considering the usefulness of a family of four-qubit mixed
states as resources for two-qubit teleportation. Our results generalize those
in [PRL, {\bf 72}, 797 (1994)].
