Akaal Bodhan () is the worship of Durga—an incarnation of Devi—in the month of Ashwin, an uncustomary time for commencement of the worship.

Etymology
Both the words Akaal and Bodhan are Bengali words that descendant from Sanskrit word akālabodhana , which are also included in many other Indian languages, such as like Bengali. 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 (Boshonto).

Akaal Bodhan
The reference to Akaal Bodhan is found in the Mahabhagavata Purana. According to this Purana, when Ravana awakened Kumbhakarna and sent him to the war, Rama was terrified. Then Brahma assured Rama and told him to worship Goddess Durga (Mahabhagavat 41.22-33) for success on the battlefield. Rama told Brahma that this was not the proper time (Akala) to worship the Goddess as this was Krishnapaksha and it is time for her sleep. Then Brahma said that on behalf of Rama, he will perform the Bodhana and the Puja. Rama agreed and appointed Brahma as the Purohita (priest) (Mahabhagavata 42.12-27). Brahma agreed to be his purohit and perform the puja from Krishna Navami till the death of the Ravana (Shukla Navami)(Mahabhagavata 43.81-97). Following the instruction of Brahma, Rama praised the Goddess by uttering Katyanai Stab (Mahabhagavata 44.1-17). Brahma himself uttered the Devi Sukta from the Vedas to please the Goddess (Mahabhagavata 45.4-19). Devi was pleased and she appeared and said that there will be a great war between Rama and Ravana between Shukla Saptami to Shukla Navami. On Saptami, she will enter into Rama's bow and arrows. During the time when Ashtami tithi will end and Navami will start in that juncture (Sandhi) she will cut Ravana's heads one after another. And finally, she assured that in the afternoon on Shukla Navami tithi, she will destroy Ravana completely (Mahabhagavata 45.20-43). 

There is another version of the story. 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 learned that the Goddess would be pleased only if she is worshipped with 108 Neel Kamal (blue lotuses). After travelling far flung places, 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. Devi took the form of Chandi to kill demon brothers Chanda and Munda. Chanda and his brother Munda could not be killed in day time or at night so the Sandhi-kshna  was chosen to kill them (i.e., the crossover period between Ashtami [the next day] and Navami [the day after]). Ravana was killed on the tenth day, 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).

See also
Durga Puja

References

Durga Puja
Shaktism