The Appearance of Lord Dattatreya:
Lord Dattatreya, the avatara of Lord Vishnu (4th avatar), His astounding form and pastimes are described very authoritatively in the Vedic literature which clears away all the myths that have grown up around this divine form of Lord Vishnu.
The great sage Atri prayed for offspring, and the Lord, being satisfied with him, promised to incarnate as his son, Dattatreya.
Sri Dattatreya is an incarnation of the Divine Trinity Brahma, Vishnu and Siva. The word Datta means "Given", Datta is called so because the divine trinities have "given" themselves in the form of a son to the sage couple Atri and Anasuya. He is the son of Atri, hence the name "Atreya".
Sri Dattatreya is usually depicted with three heads, symbolizing Brahma, Vishnu, and Shiva; past, present, and future; and the three states of consciousness: waking, dreaming, and dreamless sleep. He is portrayed sitting in meditation with his Shakti beneath the audumbara tree. In front of him is a fire pit, and around him are four dogs. These are sometimes said to symbolize the four Vedas.
According to Brahmanda-purana, Lord Dattatreya appeared during the reign of Vaivasvata Manu (who is still ruling), in the 10th yuga cycle in the Treta-yuga. Since we are currently in the 28th Kali-yuga, He appeared occurred about 80 million years ago to Atri and Anasuya,His parents.
Devi-bhagavata-purana suggests an even older date (2nd yuga cycle). The differences in dates may reflect the different appearances in the various kalpas.
The story itself, alluded to in Srimad Bhagavatam, is described in detail in the Markandeya-purana (16)
Once there was a brahmana who was cursed to die at sunrise, and his devoted wife by the power of austerity, restrained the sun from rising, upon which all activity ceased. The gods, alarmed by this, went to Lord Brahma for counsel:
Lord Brahma said: "Majesty is subdued by majesty indeed, and austerities also by austerities, O ye immortals! Hearken therefore to my advice. Through the might of the faithful wife the sun does not rise, and from his not rising loss befalls mortals and you. Hence do ye, through desire that sun should rise, propitiate Atri's faithful wife Anasuya who is rich in austerities."
Having been so instructed, they sought the help of the wife of the sage Atri. When they submitted their desires, she replied, "The might of a faithful wife may not be lost in any wise. Hence while honoring that good lady, I will liberate the day. O ye gods! that day and night may again exist, and that that good lady's own husband may not perish." At Anasuya's exhortation the brahmana's wife relented, the sun rose, and the brahmana died, but he was restored to life by the virtuous Anasuya.
Then there fell a shower of flowers, accompanied with the strains of heavenly instruments and other musical instruments. And the gods were delighted and said to Anasuya:
"Choose a boon, O blessed lady. Inasmuch thou hast accomplished a great matter for the gods, therefore the gods will grant thee a boon, O ascetic lady."
Anasuya spoke: "If ye gods headed by Brahma, being favorable, will grant me a boon, and if ye deem me worthy of a boon, then let Brahma, Vishnu, and Shiva become sons to me, and let me in company with my husband attain religious devotion, to the end that I may be delivered from affliction."
"Be it so!" exclaimed Brahma, Vishnu, Shiva and the other gods to her; and they departed, duly honoring the ascetic lady.