Some epithets are known by the Latin term epitheton necessarium because they are required to distinguish the bearers, e.g. as an alternative to ordinals after a prince's name — such as Richard the Lionheart, or Charles the Fat alongside Charles the Bald. Still the same epithet can be used repeatedly, in different spheres of life and/or joined to different names, say Alexander the Great as well as Catherine the Great.
Other epithets can easily be omitted without serious risk of confusion, and are therefore known (again in Latin) as epitheton ornans; thus the classical Roman author Virgil systematically called the armsbearer of Aeneas, his main hero, fidus Achates, the epithet being fidus, which means faithful or loyal.
In contemporary usage, epithet is also used to refer to an abusive or defamatory phrase, such as a racial epithet.
Alternatively the epithet may identify a particular and localized aspect of the god, sometimes already ancient during the classical epochs of Greece or Rome, such as a reference to the mythological place of birth or numinous presence at a specific sanctuary: sacrifice might be offered on one and the same occasion to Pythian Apollo (Apollo Pythios) and Delphic Apollo (Apollo Delios). A localizing epithet refers simply to a particular center of veneration and the cultic tradition there, as the god manifested at a particular festival, for example: Zeus Olympios, Zeus as present at Olympia, or Apollo Karneios, Apollo at the Spartan Carneian festival.
Often the epithet is the result of fusion of the Olympian divinity with an older one: Poseidon Erechtheus, Artemis Orthia, reflect intercultural equations of a divinity with an older one, that is generally considered its pendant; thus most Roman gods and goddesses, especially the Twelve Olympians, had traditional counterparts in Greek, Etruscan, and most other Mediterranean pantheons, e.g. Jupiter as head of the Olympian Gods with Zeus, but in specific cult places there may even be a different equation, based on one specific aspect of the divinity. Thus the Greek word Trismegistos "thrice grand" was first used as a Greek name for the Egyptian god of science and invention, Thot, and later as an epitheton for the Greek Hermes and, finally, the fully equated Roman Mercurius (Mercury; both were also messenger of the gods). Among the Greeks, T. H. Price notesthe nurturing power of Kourotrophos might be invoked in sacrifices and recorded in inscription, without specifically identifying Hera or Demeter.
Some epithets were applied to several deities of a same pantheon, rather accidentally if they had a common characteristic, or deliberately emphasizing their blood- or other ties; thus in pagan Rome, several divinities (including demi-gods, heroes) were given the epitheton Comes as companion of another (usually major) divinity. An epithet can even be meant for collective use, e.g. in Latin pilleati 'the felt hat-wearers' for the brothers Castor and Pollux. Some epithets resist explanation.
Similar practices still exist in Catholic and Orthodox Christianity in the veneration of Christ and, mainly, of the saints. "Our Lady of Lourdes" is essentially periphrasis, unless some aspect of the Virgin were being invoked.
Indeed while these differ from official titles as they don't express any legal status, epitheta have been awarded and adopted (though the official procedure may provide for the formal decision to be issued by another institution, such as a legislative assembly) by statesmen in power for fairly formal use, not unsimilar in purpose to various sinecures, knighthoods or peerage-type titles in post-feudal societies: they confer prestige without any legal authority, so essentially a matter of image or even propaganda, aimed at a domestic and/or foreign target audience. Examples of such epithets are the various traditions of victory titles (see there) awarded to meritous generals and rulers since Antiquity, and the epithets awarded to entire units, e.g. such adjectives as 'Fidelis' 'loyal' to various Roman legions.