Dasara festival is celebrated to remember the victory of the god Sri Rama over demon king Ravana.
Dasarth asked Rama to leave for forest for 14 years just before he was to be named as the king of Ayodhya. This request was a result of the boon that was given to Kaikeyi ( the third wife of Dasarath) by Dusserath. Kaikeyi asked that Rama should go to live in forest for 14 years and her son Bharat has to be made the king.
So Lord Rama left for the forest. He was followed by his wife Sita and brother Laxmana. 13years went normally when all of a sudden all the hell broke. Laxmana had cut the nose and ears of Shurphanakha, the sister of Ravana. On seeing Shurpanakhas state Ravana was raged and wanted to take revenge. He came to the place where Lord Rama, Sita and Laxmana were staying. Seeing Sita he was moved and by hook or cook wanted Sita.
By tricking Rama and Laxmana to chase a deer, Ravana abducted Sita. Sita was taken to Lanka the kingdom of Ravana and held captive there. Ravana was afraid to touch Sita, as there was a curse that he would die instantly if he touched a lady without her consent.
Rama in search of Sita, made friends with Sugriva by killing the rival Vali the elder brother of Sugriva. Hanuman was the man who made Sugriva and Rama friends. With the help of Hanuman and other monkeys Rama was able to find where Sita was. Rama built a bridge to reach Lanka and attacked Lanka with the army of monkeys. In the war Rama killed lot of demon warriors. And finally there was a huge battle between Rama and Ravana. However Rama tried Ravana was not getting killed.
Rama performed the Chandi Homam and pleased Goddess Durga gave the secret to kill Ravana. With the help Rama killed Ravana on Dasara and hence the day is known as Vijaya Dasami. Devotees would perform Chandi Homam on Dasara and if possible would perform the Aditya Homam.