What is Freud’s theory of Personality Development, and is it still relevant today?

Freud thought that personality developed during one's childhood. This occurred as a child moved through five stages, which he called "psychosexual stages" of development. The stages—oral, anal, phallic, latent, and genital—each involved the ability to satisfy particular needs associated with a certain part of the body.
Because unlimited satisfaction of desires is not conducive to a well-adjusted life, Freud argued that the id, which pursued desire in an animalistic way, had to be controlled by the ego and superego. The extent to which these different parts of one's personality developed determined how well-adjusted a person was. Progression through each stage involved a conflict that had to be resolved in order to develop in a healthy way, and mental illness, such as it was understood, resulted from the failure to resolve these conflicts. For a boy, resolving the strong feelings of attachment to one's mother, for example, was believed to be very important to growth and development. The resolution to the conflict that this entailed was for the boy to emulate his father's masculine behaviors. 
As for the relevance of Freud's theory, most of it has been supplanted by subsequent research and theory. Freud's assumptions about gender, in particular, were very much a product of his own time, and do not represent the conclusions of modern science. But his theory continues to provide a valid framework upon which subsequent research into psychological development has been based.