Victory of God Rama over Ravana
Rama had performed "Chandi Homa" and invoked the blessings of Durga, who blessed Rama with secret knowledge of the way to kill Ravana. On the day of Ashvin Shukla Dashami, Rama's party found Sita and defeated Ravana. Thus it is termed as Vijaya Dashami. Based on the inferences from Valmiki’s Ramayana, Kalidas’s Raghuvans, Tulsidas’s Ram Charit Manas, and Keshavdas's Ram Chandra Yas Chandrika as well as common perception in India, Rama, Sita, and of Lakshmana returned to Ayodhya on the 30th day of Ashvin (19–20 days after Vijayadashmi). To mark the return of Lord Rama, in the evening, the residents of Ayodhya lit their city with millions of earthen lamps (called Deepak). Since then, this day is celebrated in India as Deepawali or Diwali.
Many people perform "Aditya Homa" as a "Shanti Yagna" and recite Sundara Kanda of Srimad Ramayana for 5 days. These Yagna performances are thought to create powerful agents in the atmosphere surrounding the house that will keep the household environment clean and healthy. These rituals are intended to rid the household of the ten bad qualities, which are represented by 10 heads of Ravana as follows:
Kama vasana (Lust)
Mada (Over Pride)
Some householders perform Yagnas thrice daily along with Sandhya Vandana, which is also called Aahavaneeya Agni, Grahapatya Agni or Dakshina Agni. In addition, the Aditya Homa is performed with the Maha Surya Mantras and the Aruna Prapathaka of the Yajurveda. These mantras are believed to keep the heart, brain and digestive functions in balance in the absence of adequate sunlight in the winter months.
Victory of Durga Mata over Mahishasura
Some of the demons, or Asuras, were very powerful and ambitious and continually tried to defeat the Devas, or Gods, and capture Heaven. One Asura, Mahishasura, in the form of a buffalo, grew very powerful and created havoc on the earth. Under his leadership, the Asuras defeated the Devas. The world was crushed under Mahishasura's tyranny, the Devas joined their energies into Shakti, a single mass of incandescent energy, to kill Mahishasura.
A very powerful band of lightning emerged from the mouths of Brahma, Vishnu and Shiva and a young, beautiful female virgin with ten hands appeared. All the Gods gave their special weapons to her. This Shakti coalesced to form the goddess Durga. Riding on a lion, who assisted her, Durga fought Mahishasura. The battle raged for nine days and nights. Finally on the tenth day of Ashvin shukla paksha, Mahishasura was defeated and killed by Durga.
Hence Dasha-Hara is also known as Navratri or Durgotsav and is a celebration of Durga's victory. Durga, as Consort of Lord Shiva, represents two forms of female energy – one mild and protective and the other fierce and destructive.