Introduction
Tarakaksha is one of three Asura brothers, along with Kamalaksha and Vidyunmali, who ruled the three flying cities known together as Tripura, "the three cities", which were destroyed by Shiva in a single, celebrated act.
This article follows the Puranic tradition. Where a detail differs between texts, the article says so.
The three cities
The three brothers, sons of Taraka in most versions, performed austerities and were granted three magnificent flying cities, of gold, silver and iron, that moved through the sky and could only be destroyed by a single arrow fired when all three aligned in one line, an event that happened only once in many years.
Destruction
Shiva, as Tripurantaka, "destroyer of Tripura", mounted a chariot built by the gods, with the earth as his chariot, the sun and moon as its wheels, and Vishnu himself as the arrow, and waited for the alignment. When it came, he fired a single arrow that destroyed all three cities and their inhabitants at once, including Tarakaksha and his brothers.
Meaning
The Tripura Samhara is one of Shiva's major mythic acts, celebrated in temple sculpture and hymn across India, symbolising the destruction of the three-fold bondage (or the three cities of ego, illusion and desire, in later allegorical readings) by divine power.