Akalbodhan is the worship of Devi Durga in the month of Ashwin which is an uncustomary time for commencement of the worship. It is called so since the period of the worship of Goddess Durga differs from the conventional period or during the spring season (Basanta).
According to Ramayana, Rama was engaged in a fierce battle with Ravana and before going into the battle he wanted to seek blessings of victory from Goddess Durga as she was the Shakti. Rama therefore held the puja to wake up the goddess Durga during autumn worshiping Durga in 'Akal' (wrong time). From that period this puja is called 'Akalbodhan'.
According to rituals, Devi Durga is worship with the customary 108 Neel Kamals (blue lotuses). During the puja, Rama could only manage to gather 107 of them. Rama offered one of his eyes as a substitute of 108th lotus to make up the numbers and pleased with his devotion Devi Durga blessed him. Rama started his battle with Ravana on the day of Maha Saptami and Ravana was killed in between Ashtami and Navami or the 'Shandhikshan' and was cremated on the day of Dashami. Therefore the four days of puja that we celebrate marks the end in the triumph of the good over evil.
Akalbodhan is also described in the 'Kritivaas' writing. During the war between Rama and Ravana, both were equally handled by each other. After sometime, Ravana is in prayer of Rama to get the victory of this battle which made Rama surprised. Rama's first and foremost duty was to protect or to save his disciple. After that moment, Rama suffered from indecision. By seeing these, all the gods and goddesses became sorrowful and started planning to inspire Rama to kill Ravana and finally it was decided that Devi Saraswati would be on the tongue of Ravana and she will make him to say some evil words so that Rama could continue with the battle. Rama started the battle but got frustrated again and he suffered a lot for his Sita. After that all the gods felt the pain of him and decided to celebrate a Durga Puja for the intention to kill Ravana.