Brandy is distilled liquor made from grapes or other fruit like cherries or apples (e.g. Calvados). Armagnac, grappa, and pisco are all forms of brandy.
The manhattan is related to the old fashioned, a classic cocktail made with liquor plus sugar or simple syrup. The manhattan subs sweet vermouth for sugar.
A dry manhattan uses dry vermouth (left). The classic has sweet (right). The perfect manhattan (middle) uses both. Taste test all three to find your favorite.