Dash Mahavidyas:
Mahakaali:
Mahakaali is the Aadi Vidya, she takes different avatars to accomplish various tasks for Devas. In Markandeya purana, at the end of the Kalpa, when Vishnuji on his Shesh Shaiyya in ksheer Sagar was in yoga nidra, at that time 2 demons Madhu & Kaitabh, wanted to kill Brhmaji, who called Vishnuji but was not able to bring him out of his yog nidra. Brhmaji worshipped Devi Shakti, who appeared in form of Mahakaali.
In Markandeya Purana, while fighting with the demon Chanda, due to anger Devi's face turned black, from her forhead Kalika appeared.
Valmiki has written Ramayan, he has also written, Gupta Ramayan or Rdbhut Ramayan, in that while fighting with 1000 headed Ravan, Ram turned inconscious. Sita saw, this got angry turns black, as Mahakaali kills this Ravan, who was 10 faced Ravan's brother. Then Ram worships her with her sahastra naam.
Mahakaali, DakshinKaali, Bhadrakaali, Smashankaali, Guhyakaali 7 Rakshakaali are various names of this Prakruti roopa Devi. Her Shiv is Mahakaal.
Tara:
In Vedas Sun is considered as main support of the world. Sun was considered the first Rudra. He was Ugra, his Shakti is called UgraTara. She rules centre of the World. After pralaya, when the world becomes smashan, becomes Shav, she stands on the heart of this Shav.
From 12 in the night till Sunrise Mahakaali rules the Universe, after that Maa Tara rules. When Mahakaali took blue coloured form, she was called NeelTara. this form was to kill the demon Hayagreeva. She can be pleased faster, is always eager to help her devotees. Her Shiv is Akshobhya.
Tripursundari or Shodashi:
Devi Aadya Shakti appears in her Dark & Fair form. She had appeared from Goddess Parashakti, created the Trimurti, was worshipped by them that's why she is called Tripura. She is Shakti of calm Shiv. panchvaktra Shiv has 5 heads, 10 hands, 3 eyes. His shakti is Shodashi. She is also called Brhma-vidya, Shri-vidya.
Once Agastya rishi saw humans living in bad condition, he was compassionate towards them, he stopped at Kanchipuram, worshipped Mahavishnu, who was pleased with his penabnce gave him her idol.To explain the importance of the Devi Shodashi, one of his ansh Hayagreeva was appointed, who explained him Devi Lalita Tripursudari's mahatmya. She gives Dharma, arth, Kaam, Moksh to her devotees.
Bhuvaneshwari:
Shodashi was instrumental in creation of the world, when the same Shakti rulls, the 3 bhuvans she is called Bhuvaneshwari. She has created Brhma, Vishnu &Shiv. She was present before the creation, when the Universe was Shunya (zero) that's why she is called Godess of Shunya Mhavidya. Her Shiv is Sadashiv.
Tripur Bhairavi:
She blesses her devotees with meditation, penance, gyan. She is worshipped in the 3rd adhyaya of Durga-saptashati. Bhairav is her Shiv.
Chinnamasta:
Once Adya Bhavani, went to take bath at a river with her friends Jaya & Vijaya. At that time she thought of creatinon of the world, she was in deep thoughts about the creation, her friends told her they are hungry. She told them to wait, again got engrossed in her thoughts. Jaya & Vijaya reminded her once again. The compassionat Goddess cut her own head with her sword, took that head in her own hand.Her friends & her own head drank the blood flowing out of her body.
Once while fighting with Shumbh & Nishumbh, a Demon called Raktabeeja, had a boon that every drop of his blood, one more Demon will appear. To finish him, this Goddess drank all his blood so a new demon will not be created. She destroys the world in the end.
She is also called Prachanda Chandika. Her Shiv is Kabandh.
Dhumavati:
Goddess Bhagavati faces many problems & obstacles in the creation of the world. Due to that she starts looking old, like a widow. She destroys all the negative qualities of her devotee & guides him spiritually.
The smoke that came out from burning Sati's body is believed to be her form.
As per Markandey puran, she is not widow by unmarried. Once in a war she did a pratigya, whoever will defeat her in war, will become her husband. She is still undeafeted.
Bagalamukhi:
Once in Satyuga there was a very big cyclone. Vishnuji got worried about the expected destruction it can create. He could not find any way out, so at Saurashtra, near Hridra lake he did penance of Goddess Bhavani, as a result Goddess appeared as Bagalamukhi, she stopped the cyclone. She is also called the Brhmastra Vidya.
Matangi:
She is the Goddess of Vidya. She appeared as the result of rishi Matang's severe penance. She is worshipped for happiness & a good marriage life.
Kamala:
Like Goddess Lakshmiji she gives wealth, prosperity. She is Sadashiv's Shakti.