There are 2 major reasons to celebrate Dussehra:
The name is derived from Sanskrit “Dasha-hara” means remover of ten referring to Lord Rama's defeat of the ten-headed demon king Ravana. On this day, Lord Ram 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 day also marks the victory of Goddess Durga over the demon Mahishasur. The name Vijayadashami is also derived from the Sanskrit words "Vijaya-dashmi" meaning the victory on the dashmi (Dashmi being the tenth lunar day of the Hindu calendar month). 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.
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.