Bodhayana Amavasya is the no moon day or Amavasi calculated a day earlier than the normal Amavas date. This particular observance is also limited to a certain section of Vaishnava Sect. Usually, in such cases the major part of the Amavasi will fall on the previous day but it ends on the next day after sunrise.
There is a popular story associated with Bodhayana Amavasya, which is related to Sahadeva, one of the Pandava brothers. Sahadeva is believed to have had the capacity to do astrological predictions. It is said that Duryodhana before the start of the Mahabharata war approached Sahadeva and asked for an auspicious time to perform puja dedicated to Goddess Durga for success in war. Sahadeva suggested the next Amavasi day.
But the other Pandava brothers were unhappy with this gesture of Sahadeva. But Sahadeva said that he will not cheat anyone as it is against his Dharma. Then Krishna is believed to have helped the Pandavas by moving the Amavasya a day earlier. And this Amavasya is known as Bodhayna Amavasi.