The legend behind her name goes thus: Once upon a time, there was a great sage called Kata, who had a son named Katya. Kata was very famous and renowned in the lineage of saints. He underwent long austerities and penance in order to receive the grace of the Mother Goddess. He wished to have a daughter in the form of a goddess. According to his wish and desire the Mother Goddess granted his request. Katyayani was born to Kata as an avatar of Durga.
Katyayani rides atop a majestic lion and is (typically) depicted with four arms. In Her left hands She holds a sword and a lotus flower (symbolising blooming above the murk to achieve enlightenment); Her right hands are positioned in Abhaya( indicates for the devotee not to fear and is a symbol of protection) and Varada (symbolises the giving of blessings) mudras. In the Vamana Purana, She is depicted in Her fiercest form during the battle with the demon Mahishasura, whom She slays.
Katyayani rules over the planet Jupiter. Her chakra is the most important of all chakras: the Ajna (command, knowledge, wisdom) chakra, or Third Eye. This is where the bridge between our human potential crosses into Divine Consciousness. Hindus also believe that spiritual energy from the outside world enters the conscious via this gateway chakra, and so they take great care in protecting this chakra through religious markings (made with holy ash, vermillion, etc.) on the third eye.
She is commonly worshipped by unmarried women looking for a husband, and if a lady’s marriage is experiencing any delays Katyayani is often worshipped in order to remove the obstacles that are preventing the marriage from taking place.
Her mantra is:
Chandrahaasojjval Karaa Shaardoolvarvaahanaa
Katyayani Shubham Dadyaad Devi Daanavghatini
Which in English means:
Devi Katyayani, who holds Chandrahaas Sword and other weapons in her ten hands, rides on Lion, and destroying demons, be propitious to me.
(Note: She is most commonly depicted with four hands, but some Images do depict Her with eight, ten, and even eighteen hands!)
For purposes of pronunciation, I have included a link to the spoken mantra (The mantra itself lasts for the first 35 seconds of this video, after which a speaker gives a lecture in Hindi).