The term double star (or visual double) refers to two stars that are very close to each other across the celestial sphere. They may constitute a binary star or they may be separated and not gravitationally bound, e.g., one of the pair being parsecs further from Earth than the other. Sometimes the term double star is used specifically intending to mean this latter case, i.e., not a binary star. The term is also sometimes used specifically to mean specifically the former case, i.e., gravitationally bound, so it is best to check the intended meaning.
With the advent of the telescope, more stars could be seen to be double. For centuries, catalogs have been created specifically to record them. The were earlier of particular interest as asterisms (for guide stars, navigation, etc.) as well as the evidence they offered on the nature of stars: noting a rotation that suggested the two are in orbit, and in other cases, noting a change in the angle between them, indicating proper motion and its implications.
Typically astronomers gave such a pair a single designation even thought it is two stars, and the individuals are commonly distinguished by adding "A" or "B", and sometimes adding "AB" if specifically referring to the pair. (More recently, extra-solar planets have been indicated by "b", "c", etc., generally using capital letters for stars and lower case for planets.) The designations used for these double stars often follow a pattern based upon the early catalogs: an indicator of the catalog (presumably that which first recorded it, i.e., the discoverer) followed by that catalog's identification number. Recent catalogs often list such designators for cross-referencing.