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.