Mythology behind Durvasa's Birth
As per the chapter 44 of the Brahmananda Purana Lord Brahma and Shiva once got into a major quarrel. Shiva's anger was so violent that every Deva was complaining of him and was scared. Shiva realised his anger to be intolerable and decided to deposit this anger into Anasuya, the wife of sage Atri. When a child was born of this couple out of the portion that Shiva deposited to Anasuya, he was named Durvasa and was the carrier of irascible nature of Lord Shiva.
The Bhagavata Purana gives a somewhat different account of Durvasa's birth. According to this Atri performed severe penance to propitiate the Holy Trinity in order to obtain a son by Anasuya who would be just like him. The trio were pleased with him and blessed the sage that portions of themselves would be born as his sons. Later on, Anasuya bore Soma (Brahma's incarnation), Dattatreya (Vishnu's), and Durvasa (Shiva's).
Legends of Durvasa
There are many tales related to various curses given by Durvasa. Lord Indra was once cursed by Durvasa as it is stated in the Vishnu, Vayu, and Padma Puranas that was the indirect reason for the churning of the ocean. The Bhagavata and Agni Puranas also mention Durvasa's involvement in the episode.
Durvasa's curse for Ambarisha, the great devotee of Vishnu is also famous. In the Abhij¤anashakuntala, written by Kalidasa Durvasa cursed Shakuntala for ignoring him. The sage's curse came true of course.
In the UttaraKanda of Valmiki's Ramayana it is said that how to please the angry Durvasa Lord Rama had to violet his word and how Lakshmana had to die.
Durvasa was well known for giving boons too especially to them who satisfied his needs. He used to be pleased particularly when he had been served well as an honoured guest. Durvasa was gratified by the service of Kuntiwho took care with his unreasonable request. As a result before departing, he rewarded Kunti by teaching her Atharva Veda Mantras which enabled her to invoke any god of her choice to beget children by them.
In Mahabharata Draupadi's modesty was saved by this sage who provided the unending stream of cloth when Dushasanawas trying to strip her.
He is commonly considered as desiring to enjoy others' hospitality and service, and becoming exceedingly angry when hosts fail to please him as a guest. Conversely, hosts who serve him well are often blessed by him.