Suppose Alice and Bob jointly possess a pure state, \|ψ>. Using local operations on their respective systems and classical communication it may be possible for Alice and Bob to transform \|ψ> into another joint state \|φ>. 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.