Shiva is the chief god of Shaivism, one of the two main branches of Hinduism today (the other being Vaishnavism). His abode is called Kailasa. His holy mount (called vahana in Sanskrit) is Nandi, the Bull. His attendant is named Bhadra. He is usually represented by a phallus, the Shiva linga (or lingam).
Shiva's consort is Devi, a goddess who comes in many different forms, one of whom is Kali, the goddess of death. Parvati, a more pacific form of Devi is also popular. Shiva also married Sati, daughter of Daksha, who forbade the marriage. Sati disobeyed her father and Daksha held a sacrifice to Vishnu, but did not invite Shiva. In disgust, Sati sacrificed herself in the same fire Daksha used in his sacrifice. Shiva crashed the party, angry at the death of his wife, and killed many of the guests, as well as decapitating Daksha, though he later replaced his head with that of a goat. Shiva created the monster Virabhadra during his quarrel with Daksha.
Shiva gave Parashurama his axe.
Shiva and Parvati are the parents of Karttikeya and Ganesha, the elephant-headed god of wisdom. He acquired his head due to the actions of Shiva, who decapitated him because Ganesha refused to allow him to enter the bath while Parvati was bathing. Shiva had to give him the new head to placate his wife. In another version, Parvati showed the child off to Shiva, whose face burned his head to ashes, which Brahma told Shiva to replace with the first head he could find, an elephant.
According to the foundational myth of Kalism, Kali came into existence when Shiva looked into himself; she is his mirror image. Kali is often depicted standing on top of Shiva's dead and lifeless body. Shiva is shown dead to signify his passiveness in relation to Kali.
As Nataraja, Shiva is the God of Dance.
Traditionally, unlike Vishnu, Shiva does not have any avatars. However, several persons have been claimed as avatars of him, such as Shankara. Some people consider Hanuman to be an avatar of Shiva.
Nayanars (or Nayanmars), saints from Southern India, were mostly responsible for development of Shiva cult in the Middle Ages.
The important Shaivite cults were Kashmir Shavaites from Northern India, Lingayats and Virasaivas from Southern India. Saiva Siddhanta is a major Shaivite theory developed in Southern India.
This is not complete yet. More details on Shaivite texts/schools needed.
See also Ardhanari