How was Balarama born? What is the story behind that?

Balarama is son of Vasudeva. The evil king Kansa, brother of Devaki, was intent upon killing the children of his sister. Because of a Akashvani (prediction) that he would die at the hands of her eighth child.
Evil demon Kansa had already killed the first six children of Devaki by smashing the newborns on a stone.

Vishnu incarnated as Balarama when Indra and the other gods came to Lord Vishnu and asked for release from the king Kansa.
“ପୂର୍ଣ୍ଣଚନ୍ଦ୍ର ହେଲେ ପନ୍ଦର କଳା,
ତହିଁ ରୁ ଅଧିକ ନନ୍ଦ ବଳା” _ଜଗନ୍ନାଥ ଦାସ
Vishnu took one Black and one White hair from his head and promised that these two would come to fight against the demons. This way, first Balrama (the white hair) and then Krishna (the black hair) incarnated in the womb of Devaki. When Balarama appeared as the /Seventh child in the womb of Devaki/, she could understand that this was a divine child and this made her all the more concerned about His safety. Even Kansa could sense His potency and he became fearful, thinking he may have been tricked by the prophecy that he will be slain only by the eight child of Devaki. At this time Lord Vishnu instructed Yogamaya, His internal potency, to transfer the unborn child from the womb of Devaki to that of Rohini, one of the other wives of Vasudeva. In this way Balarama was born in Gokul under the protection of Nanda Maharaja.

Garga Muni the Kulguru (family priest) of the Yadu-Bansh (dynasty) revealed to Rohini that the child she was carrying was indeed that of her husband Vasudeva. He and Krishna grew up together, and he took part in many of Krishna’s boyish freaks and adventures.

Since He was forcibly attracted from the womb of Devaki to that of Rohini, So He also be called Sankarshana (one who was dragged away).
Another view is held as to the origin of Balarama. According to this, He was Second incarnation of the great serpent Seshanag and when He died, the serpent is said to have issued from his mouth.

Lord Balarama is the eternal companion of Sri Krishna. He came in Satya Yug as Lakshmana with Rama and later in Kali Yug as Nityananda Prabhu with Chaitanya Mahaprabhu.

He is the source of the entire spiritual world and is the Adi-Guru, the original spiritual master.

In the 10th chapter of the Shrimad Bhagavata Purana describes it as follows:

At the time of the name-giving ceremony Garga Muni think to named the child Rama, one who gives all pleasures but for the immense Strength of the child, he named him as Balarama or Balabhadra (‘Bala‘ means ‘Strength‘ and ‘Bhadra‘ means ‘Descent‘ ).