What does alpha dog mean?

It means the leader of a wolf pack. Usually the alpha (leader) of a wolf pack is a male so they just call it an alpha male. Sometimes though, rarely, a she-wolf is known to be the alpha.

An alpha male can also mean the male wolf who can produce pups, but that is usually the leader (the alpha male). I also think that the alpha she-wolf is the she-wolf who can bear pups, or the mate of the alpha (leader).