Durga - The Invincible one

Shakti the divine feminine half of Shiva has two aspects: the benevolent and the fierce. She is worshiped in various forms of the above two aspects. The Benevolent forms are Uma, Gauri, Parvati, Jagatmatha. And the Fierce forms are Durga, Kali, Chandi, Bhairavi. We find mentions of Durga, the invincible one, in the Vedic literature, Mahabharatha and Ramayana.

According to Shiva Purana and Devi Mahatmiya, Durga first appeared, as a beautiful warrior seated on a tiger. This form of Shakti was created from the combined anger of the Gods and Lord Vishnu, for the annihilation of the demon Mahishasura ( buffalo demon ), who became invincible through great penance and unleashed terror on earth and defeated the Gods. Durga had eighteen arms and each hand held a different weapon, given by the Gods. She mounted a ferocious lion, fought with Mahishasura and vanquished him. So she is also known as Mahishasuramardhini. Symbolically Durga epitomises the victory of good over evil.

Another version of the story is that Shiva helped Parvathi to realise that she is Adishakti (the pure energy of the Universe). In times of great distress and helplessness, Adishakthi manifested in the Fierce forms to eliminate the evil: as Durga to kill Mahishasura and as Kali ( the Black one) to kill Raktabija. Thus Durga as the mother of Universe, provides humans with the opportunity for a peaceful life and Salvation.

Durga Puja is celebrated in India, Nepal, Bangladesh and all over the world. In India, generally it is celebrated from the sixth to the tenth day of the waning moon in the month of Ashvin (September - October). In Nepal the festival is known a Dashain and celebrated for ten days. Durga is worshiped in her ten forms, one form per day. The ten forms are Kushmanda, Chandraghanta, Brahmacharini, Shailaputri, Skandamatha, Katyayani, Kalarathri, Mahagauri, Mahakali and Durga. 

In North India the festival is known as Dussehra. Both Dussehra and Dashain mean 'victory on the tenth day'. In the North Dussehra is celebrated as the victory of Ram over Ravana. Gigantic straw effigies of Ravana are burnt on that day. Elsewhere,  Durga is worshiped on the preceding nine days (Navarathri) and her victory is celebrated on the tenth day ( Vijaya Dasami ). As in Dashain different forms are worshipped each day. In West Bengal, huge Durga Pandals are put up. 

In Telengana, Durga is celebrated as Bathukamma - a flower festival during Navarathri. Bathukamma is the cultural icon of Telengana. 

Durga Puja, also, commemorates the annual visit of the Goddess with her children ( Ganesh and Karthikeya) to her parents' home (Himalaya) and leaving on the tenth day to be reunited with Lord Shiva. The immersion of the Durga idols on the tenth day symbolizes this. 

Now, in the Northern States of India Durga is worshiped as a gentle bride epitomizing family unity and in the Southern states and elsewhere as a warrior. 

A part of the Durga Sloka / Devi Mantra : 

Ya Devi Sarvabhuteshu Matri rupena samsthita 
Ya Devi Sarvabhuteshu Shakti rupena samsthita 
Ya Devi Sarvabhuteshu Shanti rupena samsthita 
Namastasyai Namastasyai Namastasyai Namoh Namah 
Meaning: 
The Goddess who is omnipresent as the personification of the Universal mother
The Goddess who is omnipresent as the embodiment of Power 
The Goddess who is omnipresent as the symbol of Peace 
I bow to her, I bow to her, I bow to her.