Tripurari -- The Enemy Of The Three Cities.
The three sons of Tarakasura who were called Taraksha, Kamalaksha, and Vidyunmalin, obtained a boon from Brahmaji by virtue of their arduous tapa that they would posess three indestructible flying cities, one of gold, one of silver, and the thrd of iron. They used these cities to terrorize the worlds. They also asked for the boon of immortality for themselves and their demonic army. But all things that are born must die. Even the mighty Devas for whom even the longest human life seems like the blink of an eye, have a similiarly short lifespan on the scale of Brahman. So they were denied that wish. Instead they asked "let us only be killed by Mahadeva." They knew Shankara Bhagawan was ever immersed in dhyana and would not be a threat to them. For a thousand years the Asuras continued their terrible reign and because of the boons the Devas were unable to do anything about it. Weeping and lamenting they went to Mount Kailasa to the abode of Lord Shiva. One name of Shiva Bhagawan is Rudra - one who crys or howls. Because he is always indignant at adharma in the world. He is also called Ashutosha - He who wipes away tears because He is always ready to comfort His bhaktas. Seeing the distress of the Devas and Rshis and the wicked acts of the Asuras, He roused Himself from his yoga and prepared for battle. As he strung his arrow, the three cities came together and with one shot he pierced and destroyed them all .
The three cities are the three gunas - sattva, rajas, and tamas. Brahman is nirguna, beyond even sattva. The pauranika katha emphasizes this even more when it mentions the three cities were built by Maya (personified as the architect of the Asuras.)
Some additional discussion and notes:
> One of the most important nams of Mataji is Tripura, or Tripurasundari. I
> would like to know what connection that has with this story.
Tripuraari is also referred to as puraari, purahara, and just tripura. Tripurasundari means the wife of tripuraari.
Tripurasundari ashhtakam (composed by Shankaracaarya) refers to Her as trilocana-kutumbiniim tripurasundariim aashraye [ I take refuge in tripurasundarii who is the spouse of the three-eyed one]
Soundarya lahari (verse 7) calls her pura-mathitu-raho purushikaa [the I-ness of the one who destroyed the pura-s].
The word Tripura itself has been taken to mean any of the triads. The story of the destruction of the three pura-s is mentioned in shivaananda lahari (verse 3 ). There , tripurahara has been interpreted as one who destroys the sthuula, suukshhma, and kaaraNa shariira-s. and gives videha mukti.
Lalitaa sahasranaama lists a number of names like tripureshii , tripuraambikaa, tripuramaalini, mahaa-tripura sundarii,etc In all these cases, the word tripura is interpreted in terms of Her being mother of different triads like jagrat-svapna- sushupti ; bhuloka-bhuvarloka-suvarloka; knower-known-knowledge ; and so on.
Tripura-sundari means the spouse of tripuraari. In the muuka-panca-shati stotram She is addressed as pura-hara-jaaye -- jaaya means wife, pura-hara is the same as (tri) puraari or tripuraantaka.
OM TripurAraye namah