Brandy is a distilled spirit, made from fermented fruit. The simple term Brandy, when used by itself, always refers to a spirit that has been distilled from grape wine.
Cognac is a special designation for fine brandy, and can only be produced in the Cognac region of France. Brandy on the other hand can be made anywhere. Thus all Cognac is brandy, but all brandy is not Cognac.
Armagnac is made in much the same way as Cognac, the main difference between the two is that Armagnac is produced in the Department of Gers, southeast of Bordeaux. Another difference is that while Cognac is typically distilled three times, Armagnac is only distilled once. There are differences in taste and character between these two fine brandies. Armagnac has a drier, harder taste than cognac - and for this reason is preffered by some connoisseurs. Other forms of brandy include Marc (pronounced "Mar"), Grappa, Calvados, Pisco, Metaxa, Ouzo, Raki, Kirsch, Schwartzwalder, and various other local varieties.