Aššūr is the name of a god and an ancient city, the capital of the empire of Assyria (from the 14th–9th century BCE; later the capital was moved to Nineveh). The name Assyria, and probably also Syria, derives from him. His temple in the capital was the cultic center of the empire, at least from the perspective of its rulers.
His name is spelled phonetically as aš-šur (𒀸𒋩). When the god rather than the capital or empire is meant, the determiner 𒀭 DINGIR (‘god’, abbreviated ᵈ) is added (𒀭𒀸𒋩). For the sake of brevity, the last sign may be dropped (𒀭𒀸 ᵈAŠ).
Alternatively, Aššūr’s name can be spelled 𒀭𒊹 (AN.ŠAR₂), a spelling that reflects a conflation with Anšar, an otherwise obscure god who features prominently in the Enūma Eliš. In this spelling, 𒀭 functions to indicate the first syllable (aš), and is not repeated to also serve as the determiner DINGIR. (Similarly, the god An is spelled as 𒀭, not 𒀭𒀭.)
Aššūr was exalted as the highest god in the Assyrian empire, and continued to be revered after its fall, although to my knowledge, there is no clear evidence that this tradition survived past the end of cuneiform literacy and into the 1st millennium CE. He is mentioned in the 6th-century CE philosopher Damascius as Assōros (gr. Ἀσσωρός) – see here –, but he had his information from a much earlier Greek account of the Enūma Eliš, not from a surviving tradition of worship. (Although of course absence of evidence is only evidence, not proof of absence.)
2 The family of the god Aššūr
According to Wiebke Meinhold (“Die Familie des Gottes Aššur”, in: Marti Lionel, ed., La famille dans le Proche-Orient ancien: réalités, symbolismes, et images, 2014), sources from the late 3rd and early 2nd millennium BCE do not mention any family relations of Aššūr.
Later, Aššūr was said to be the spouse of Mullissu, another Assyrian deity (equated with Ninlil), and by equation with Enlil, Ninurta is sometimes called his son, and Nusku his minister (sukkal). Unlike Enlil, however, he is not considered the son of Anu. In the Assyrian recension of the Enūma Eliš, he is rather identified with Anšar (see above), the father of Anu and grandfather of Enlil. But by this account, he still has parents, whereas other Assyrian texts go further, calling him bānû ramānīšu, “creator of himself”:
“(Aššūr is the) king of all the gods, the creator of himself, the father of the great gods, whose figure was exalted in the Abyss, king of heaven and earth, lord of all the gods, the progenitor of the Igigi and the Anunnaki, the builder of the roof of heaven and the basement of earth, the maker of all the regions, living in the [pur]e starlit heave[ns], the foremost god, the one who decrees the destinies” (SAA 12 086, off-site link).
In the Neo-Assyrian period, when a temple for Aššūr was built in Nineveh on the model of that in the city of Aššūr, the Ninevites began to treat Ištar-of-Nineveh as his wife, equating her with Mullissu, while in the former capital, they were seen as distinct deities. In Arbela, analogously, it may have been Ištar-of-Arbela who was considered his spouse, and she again seems to have been equated with Mullissu. Outside of these local contexts, however, Ištar-of-Nineveh and Ištar-of-Arbela were treated as distinct deities, not just separate from each other and Mullissu, but even distinguishable from the (general or unspecified) Ištar.
The goddess Šerūˀa was sometimes considered Aššūr’s daughter, sometimes his spouse (but subordinate to his main wife Mullissu). Ištar-of-Arbela too could be seen as his daughter.