There are three main stories in the Puranas related to the birth of Lord Shiva. Shaivites believe that Lord Shiva is Svayambhu, i.e., self-manifested, while Vaishnavites believe that Lord Vishnu created Lord Shiva.

The Story of the Birth of Lord Shiva per the Shiva Purana:

At the end of the Kalpa (4.32 billion years), there was only a single vast sheet of water. Lord Brahma saw Lord Vishnu having a yogic sleep on the bed of Shesha (a serpent deity of Hindus). He woke him up with the blow of his hand and asked him who he was. Lord Vishnu told him that he was the creator, sustainer, and destroyer of the world. It infuriated him because he believed he was the creator, sustainer, and destroyer of the world. Then they got into an argument about who was superior.

The argument turned into a furious fight, and then suddenly, a Jyotirlinga, a huge, infinite pillar of light, appeared before them. It had thousands of clusters of flames and no beginning, middle, or end. It was the source of the universe. They forgot their fight and decided to test it. Lord Brahma assumed the form of a swan and went upwards. Lord Vishnu took the form of a wild boar and went downwards.

They both traveled for one thousand years but could not find the end of the lingam. Therefore, they returned to where they started. They bowed to the lingam and wondered what it was. Then a loud sound “Om” came out of the column, and the letters “अ” “, ऊ,” and “म” (“a,” “u,” and “m”) appeared on the lingam. Above those letters, they saw Lord Shiva along with the goddess Uma. Lord Shiva told them they were both born out of him, but they had forgotten about that.

This story is known as Lingodbhava, i.e., the emergence of Linga.

(As per Shiva Purana, once Lord Shiva was rubbing nectar on his ankle, and Lord Vishnu was born).

The Story of the Birth of Lord Shiva per the Vishnu Purana:

1. At the beginning of the Kalpa, as Brahma purposed to create a son, who should be like himself, a youth of a purple complexion appeared, crying with a low cry, and running about Brahmā, when he beheld him thus afflicted, said to him, “Why do you weep?”

“Give me a name,” replied the boy.

“Rudra be your name,” rejoined the great father of all creatures: “be composed; desist from tears.” But, thus addressed, the boy still wept seven times, and Brahma, therefore, gave to him seven other denominations; and to these eight persons regions, wives, and posterity belong. Then, the eight manifestations are named Rudra, Bhava, Sharabha, Ishana, Pashupati, Bhima, Ugra, and Mahadeva, which were given to them by their great progenitor. He also assigned to them their respective stations, the sun, water, earth, air, fire, ether, the ministrant Brahman, and the moon; for these are their several forms (Book I – Chapter 8).

2. There is one more story about the birth of Lord Shiva in the Vishnu Purana, which I could not find a reference.

Lord Vishnu was the only one who survived total destruction. After some time, he started recreating the universe by giving birth to Brahma. Lord Brahma was born out of his navel. Then he created Lord Shiva from his forehead. That is why Lord Shiva is always meditating.

The Birth of Lord Shiva per the Brahma Vaivarta Purana:

According to this Purana, Lord Shiva was born from the forehead of Lord Brahma.

“The four-faced Brahma emerged out of the navel of Mahavishnu, who was reclining on the serpent bed in the ocean. Lord Siva appeared from the forehead of Brahma.”

– Krishna Janma Khanda 41.70

The Birth of Lord Shiva per the Brahma-Samhita:

Per the Brahma-Samhita, Lord Shiva was born from the space between the two eyebrows of Maha-Vishnu.

“The same Mahā-Viṣṇu created Viṣṇu from His left limb, Brahmā, the first progenitor of beings, from His right limb and, from the space between His two eyebrows, Śambhu, the divine masculine manifested halo” – Shri Brahma-Samhita 5.15.

Conclusion:

Among the stories mentioned above, the one mentioned in the Shiva Purana is the most widely accepted, and hence, Hindus believe that Lord Shiva is self-manifested and there was no birth as such of him. He just appeared from a fiery column of fire, which is Hiranyagarbha.