Sharmishtha was daughter of Daitya Vrishparva and she was friend of Devyani, daughter of Shukracharya who was Guru of Demons. Once Devyani and Sharmistha along with her friends went for a bath and Indra turning into Varuna blew away their clothes and they ran to take it and accidentally wore each other's clothes. A very bitter egoistic quarrel happened between them for wearing each other’s clothes , both trying to prove the other was unworthy to wear her cloth. Angry Sharmishtha threw Devyani in a well and left with her friends. Then, Yayati (son of Nahusha) came there and saved Devyani. Devyani fell in love and proposed him but he said he feared Shukracharya and left.
When Shukracharya saw Devyani in such a sad condition, he got angry. Devyani vowed never to enter in the Kingdom of Vrishparva. To save his Kingdom from Shukracharya’s wrath, Vrishparva made his daughter Sharmishtha as handmaiden of Devyani. Later, Devyani and Yayati got married but Shukracharya took a vow from him that he would never lay with Sharmishtha (as Kings that time had right to marry as many women he wanted and Sharmishtha was very beautiful).
While living with Devyani and seeing her happy and getting children, Sharmishtha was jealous and pitied her own condition. She proposed Yayati for marriage and he refused it saying that he had promised Shukracharya although he noticed Sharmishtha’s beauty.
Then Sharmistha gave the most ridiculous and illogical argument ever possible, she said a girl's husband is considered her friend’s husband as well. She said some more stuff and Yayati got convinced and they married each other (more like cheated Devyani and broke promise given to Shukracharya).
When Devyani got to know about the betrayal, she was beyond hurt and Shukracharya cursed young yayati with old age. He requested his son from both wives to take away his old age as he wanted to enjoy joys of life. Only Puru, his son with Sharmistha agreed and Puru became the new King. Later yayati realised the futility of having a long young life and then gave back youth to Puru and went to do penance.
Source- Adi Parva, Mahabharata