Who is Jacques Lacan and why are his theories significant?

Jaques Lacan was a French psychoanalyst who critiqued Freudianism and rose to prominence in the mid-to-late 20th century. He became controversial both for his theories and the way he conducted psychoanalysis, sometimes ending the standard 50-minute session very early and sometime letting it run very late. He thought for himself and was not prone to follow the rules.
Lacan became important to feminists because of his critique of Freud. He is famous for stating that Freudianism defines woman as a non-sequitur. Non-sequitur means "it does not follow." According to Lacan, Freud understands a woman as defined by lack—lack of a phallus. Lacan said this made no sense—a woman was never meant to have a phallus, so it doesn't follow that she "lacks" something she was never meant to have. This gave feminists an opportunity to define women in a more positive way, not merely as inferior versions of men.
Lacan was also important in emphasizing the symbolic over the literal. For example, he said men are not afraid of physical castration, as Freud argued, but that this castration anxiety symbolizes the more general fear of losing something important. He saw the unconscious as more about language and the imaginary than reality, which made him beloved both of post-structuralists (of which he was one) and literary types.
This doesn't begin to do justice to the depth and wide-ranging complexity of Lacan's work, but is a start.