For the most part, it means angry.
She has mad problems.
A MAD Movie is a Japanese fan-made video, much like an anime music video (AMV), that generally originate from the Japanese website Nico Nico Douga. MAD can also describe the Japanese AMV community, although they can be anything from audio clips, edited pictures, to wholly original creations. MADs do not necessarily even need to be related to anime, though the more popular ones typically are. The term MAD (occasionally M@D) is thought to be a reference to the name of one of the original MADs, rather than a particular acronym for anything.