Asmodeus is more ancient than the gods.
The AD&D 2nd edition book Guide to Hell (2000) details the origins and true nature of Asmodeus.
The multiverse began in chaos, and from chaos arose law. The most powerful of these were the Cosmic Serpents: Jazirian and the winged Ahriman. The two formed the undefined chaos of reality into what would be known as the Outer Planes. Each held the other's tail in their mouth, and formed an unbroken ring that defined the (then finite) scope of the planes.
But as the concepts of Good and Evil gained power, the two took sides, and fought, and each bit off the other's tail. Each drop of blood from Jazirian's tail fell and formed a couatl, but the wingless Ahriman fell into the base of the Nine Hells where he has remained ever since, biding his time until he can heal.
Ahriman took the name Asmodeus, and conducts all of his business via avatars and go-betweens. even among the gods and the lords of the Nine Hells there are none who know his true identity.
Asmodeus intentionally encourages cults of all kinds. The souls of the dead travel to the plane of their deity, or that which most closely matches their alignment if they followed no deity. But the souls of those evil peoples who reject the worship of any deity—cultists of non-divine beings, or those who reject all belief in the divine—are claimed by Asmodeus, his right as one of the Ancient Brethren who existed before there were people to believe in gods, and whose power is not reliant on faith or worship.
Asmodeus' avatar in Guide to Hell is a giant humanoid with reddish skin (not unlike many tieflings), small horns, and a finely trimmed beard. His true form is that of a serpent hundreds of miles long.
The primary source on Asmodeus in D&D 3.5 is Fiendish Codex II: Tyrants of the Nine Hells. It concurs with the AD&D description. His backstory is similar, although his relationship to the couatl god Jazirian is omitted. It is said that he bleeds perpetually from wounds.
D&D 4th edition makes Asmodeus a deity. He was already a greater deity in AD&D. Much D&D 4th edition lore is very different to earlier lore, and he was cast out by a deity he served in a manner reminiscent of the traditional story of God casting of Lucifer. Asmodeus in this edition is, as usual, described as a red-skinned humanoid with black horns and a confident, modest manner.
Asmodeus first appeared in the AD&D 1st edition Monster Manual, where he is depicted as a humanoid with small horns and a goatee beard. He is described as the most handsome, strongest and most cunning of all devils. Deities and Demigods upgrades Asmodeus to be considered a deity.
Outside of D&D 4th edition, Asmodeus has pretty consistently been described as the most powerful archdevil, and the ruler of the ninth level of Hell. The backstory with Jazirian is limited to AD&D 2nd edition (where it could be published in the year 2000 thanks to Wizards of the Coast relaxing the ban on words like "devil"), and alluded to in third edition.