Why are many alcoholic drinks refered to as "dry"?

They are mostly based on dry gin, that is, gin with no added sugar. Quite how this became known as dry, though, I have no idea.

Dry is the opposite of sweet. Dry in wines usually refers to less than 0.5% residual sugar. In cocktails it sometimes, but hardly always, indicates the inclusion of dry gin or vermouth.
e.g a very dry martini would have minimal or no vermouth.
Wines are often categorized by their sweetness or lack thereof. "Brut" wines are not sweet at all (referred to as "dry"), while "demi sec" wines are fairly sweet. Terms like "off-dry" (very slightly sweet), "extra dry" (medium sweet) and "sec" (slightly sweet) are also used. Most champagnes are brut while dessert wines are demi sec. The term usually appears on the label somewhere under the brand name.