Vijayadasami is celebrated on Ashwayuja Shuda Dasami (10th day of the waxing phase of moon in Ashwin month). Vijaya Dashami is the last day of Durga Puja and Dussehra Festival. It is observed on the day after nine-day Navratri festival. 
Dussehra, also known as ‘Vijayadashami,’ holds a special place in the hearts of Indians as one of the most significant festivals. It signifies the conclusion of the nine-day extravaganza of Navratri and the four or five-day celebration of Durga Puja.
On this day in the Treta Yug, Rama, also called Shri Ram, the seventh incarnation of Vishnu, killed the great demon Ravana who had abducted Rama's wife Sita to his kingdom of Lanka. Rama, his brother Lakshmana, their follower Hanuman and an army of monkeys fought a great battle to rescue Sita. The entire narrative is recorded in the epic Ramayana, a Hindu scripture.
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 Balmik’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 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.
Another legend has it that Goddess Durga (Mahishasuramardhini or Chamundeswari) killed Mahishasura on this day. Mahishasura – a demon with the head of a buffalo and body of human – had created havoc on the earth after defeating the demigods and unleashed the rule of Adharma. To restore Dharma, Goddess Shakti appeared as Durga. She fought Mahisha and numerous demons for nine days and on the tenth day She killed Mahisha. From then onwards the day is celebrated as Vijaya Dasami.
A further legend suggests that Arjuna in the Mahabharat during the last year of exile period took back his weapons from Shamee tree on this day and fought against Kauravas to defend the Virata kingdom. The Shami tree is worshipped with the recitation of a prayer. The meaning of the prayer is “The Shami tree cleanses sins. Its thorns are reddish in colour. It is Lord Rama's favourite tree and in such a tree Pandavas hid their arms. O Shami, Lord Rama has worshipped you. I now embark upon my journey to victory. May you make it pleasant and free from obstacles.”
Significance of Ayudha Puja
Traditionally, weapons used by soldiers are worshipped on this day. In the olden days, the Kings and their army would worship their weapons (Shastra) that shielded them from their enemies. However, in modern times, artisans, technicians, sculptors, industrialists, factory owners, mechanics etc. worship instruments, machines, equipment, tools and other items that help them in their day-to-day lives/earn their livelihood. On this day, people who own vehicles also perform puja of their Vahana to eliminate evil spirit and prevent any untoward incidents. Thus, by performing puja, devotees’ express gratitude to these inanimate but essential equipments that helps them sustain and prosper. Therefore, it is an ideal way to be grateful to anything and everything that aids support living.

Many people perform "Aditya Homa" as a "Shanti Yagna" and recite Sundara Kanda of Srimad Ramayana for nine 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 ten heads of Ravana.

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.

Brahma, Vishnu and Shiva created a 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.

Homecoming of Durga Maata
Daksha, the Lord of the Earth, and his wife Menaka, had a daughter called Sati. As a child, Sati started worshipping Lord Shiva as her would-be-husband. Lord Shiva was pleased with the Sati's worship of him and married her. Daksha was against their marriage but could not prevent it. Daksha arranged a yagna to which everyone except Lord Shiva was invited. Sati, feeling ashamed of her father's behaviour and shocked by the attitude meted towards her husband, killed herself. Lord Shiva was anguished when he discovered this. He lifted Sati's body on his shoulders and started dancing madly. As the supreme power was dancing with wrath, the world was on the verge of destruction.

Then Lord Narayana came forward as a saviour and used his Chakra to cut Sati's body into pieces. Those pieces fell from the shoulders of the dancing Shiva and scattered throughout the world. Shiva was pacified when the last piece fell from his shoulder. Lord Narayana revived Sati. The places where the pieces of Sati fell are known as the "Shakti Piths" or energy pits. Kalighat in Kolkata, Kamakshya near Guwahati and Vaishnav Devi in Jammu are three of these places.
In her next birth, Sati was born as Parvati or Shaila-Putri (First form of Durga), the daughter of Himalaya. Lord Narayana asked Shiva to forgive Daksha. Ever since, peace was restored.