  Suppose Alice and Bob jointly possess a pure state, $|\psi\ra$. Using local
operations on their respective systems and classical communication it may be
possible for Alice and Bob to transform $|\psi\ra$ into another joint state
$|\phi\ra$. This Letter gives necessary and sufficient conditions for this
process of entanglement transformation to be possible. These conditions reveal
a partial ordering on the entangled states, and connect quantum entanglement to
the algebraic theory of majorization. As a consequence, we find that there
exist essentially different types of entanglement for bipartite quantum
systems.
