Both the words Akaal and Bodhan are Sanskrit words, which are also included in many other Indian languages, such as like Bengali and Hindi. The word Akaal means untimed (kaal=time and a=not) and the word Bodhan means worship or invocation. Thus, Akaal Bodhan means worship or invocation of Durga in an uncustomary time. It is given this name since the period of this worship differs from the conventional period, which is during the spring (Basant).
In the Ramayana, Rama goes to Lanka to rescue his abducted wife, Sita, from the grip of Ravana, the king of the Demons in Lanka. Before starting for his battle with Ravana, Rama wanted the blessings of Devi Durga. He came to know that the Goddess would be pleased only if she is worshipped with 108 Neel Kamal (blue lotuses). After travelling the whole world, Rama could gather only one hundred and seven of them. He finally decided to offer one of his eyes, which resembled blue lotuses. Durga, being pleased with the devotion of Rama, appeared before him and blessed him. The battle started on the Saptami and Ravana was finally killed on the Sandhikshan (i.e., the crossover period between Ashtami [the next day] and Navami [the day after]). Ravana was cremated on Dashami. Since the period of this worship was different from the conventional period, this puja is also known as Akal-Bodhan; a worship (Bodhan) in an unconventional time (A-Kaal).