Orange rum is a type of rum flavored, either naturally or artificially, with oranges. Orange rum, like orange vodka, is typically lightly sweetened, and is bottled around 70 proof. Bacardi and Cruzan are two common producers of orange rum.
Orange rum should not be confused with orange liqueurs that have a rum base (such as Clement's fantastic Creole Shrubb). Patron's Pyrat rums are aged in barrels that have previously held orange liqueur, so they have a strong orange aroma and flavor to them -- read more about that here.