The five pandavas and the story of their birth
Preeta or Kunti an adopted daughter of KuntiBhoja was appointed by her father to serve sage Durvasa, who was known for his extreme tempers and spiritual powers.
Kunti served the sage with reverence and discipline, pleased by her devotion the sage granted her a boon, he taught her a mantra to invoke any of the heavenly gods and get a powerful son from him, and the mantra was only valid for 5 times.
Birth of Karna: Kunti delighted and very curious to test the powers of the boon invoked the Sun God, the Sun God appeared in front of her, being a maiden she was worried of the consequences of bearing a child. She requested Sun God to forgive her, that she invoked him in her girlish fickle mind. The Sun God did not agree to go without fulfilling the purpose of his visit as it was shameful for a God to do so and that other demi-Gods will talk lightly of him.
However he granted her another boon to become a virgin again after the birth of the child. Thus by the grace of the Sun God a child was born. Kunti scared of others knowing her shame placed the child in a basket and left the basket in a river. It was found by a charioteer Adhiratha and his wife Radha, they named him Vasusena and he became famous as Karna.
Birth of Pandavas: King Pandu while hunting in the forest, shot arrows and killed two deer while they were in sexual union. The deer were actually a sage and his wife who had morphed their body. For his foolish act of killing them at that intimate time the King Pandu was cursed to die if he ever embraced any woman.
Both Kunti and Madri, the queens of Pandu thus remained childless, Pandu desirous of having children was growing unhappy. Kunti revealed to him the story of her boon. Pandu delighted by the story, pleaded her to bear him a child.
Kunti and Pandu discussed whom to invoke and decided that they would like to have a son who is in righteousness (Dharma) as great as the God Yama (YamaDharma). They both meditated for few days to purify themselves and then invoked Yama; Yama appeared and blessed them with a boy as great as him in Dharma. Thus Yudhisthira was born.
After some days, Pandu and Kunti invoked Wind God desiring a son as powerful as him, blessed by the wind God Kunti gave birth to Bhima, who did mighty deeds when he grew up.
Now desiring a son who is both powerful and virtuous they invoked God Indra, the king of demi-god realm. Thus Arjuna was born.
Pandu convinced his first wife Kunti to give the mantra to his second wife Madri, so that she too can bear him a son, Madri invoked Ashwini twin-Gods, thus Nakula and Sahadeva were born.