Broadly defined, brandy is distilled fermented fruit, also known as wine. The word comes from the Dutch term brandeuwijn - which means burnt wine. Naturally, grapes are the most common base. Aged grape brandy is what you see being sipped from snifters after a Downton Abbey dinner party, or the like.
While just about every country makes their own national version brandy, Cognac - which is a brandy that can only made in the cognac region of France - is widely considered to be the best. Anytime brandy is called for in a cocktail recipe, Cognac is generally the best option.
Other prominent grape brandies include armagnac - also made in France to the south of cognac, Spanish Brandy, American brandy and Pisco - which is an unaged grape brandy made in Peru and Chile.
Brandy made from fruit other than grapes is collectively known as fruit brandy. The most notable where cocktails are concerned is apple brandy, which is made both in the United States - also called applejack, and France, where it's called Calvados. Brandy is also made from pears, apricots, cherries, peaches and plums, just to name a few. Any of these can be aged. When a brandy is unaged it’s called eau-de-vie.
Cognac, is made in the Cognac region in France and the best option anytime generic brandy is called for in a cocktail. There are three classifications of Cognac based on the minimum age in blend: V.S. ("very special") Two years, V.S.O.P. ("very superior old pale") Four years. XO ("extra old") 10 years, previously 6. Armagnac is made in the Armagnac region of France further south. It is similar, but a little more rustic.