How the date is fixed
Sharad Purnima falls on the full-moon day (Purnima) of the lunar month Ashwina. The date is the day on which that tithi is running at moonrise.
About
Sharad Purnima is the full moon of Ashwina, in October, the first full moon after the monsoon clears. It is a night observance, so the day whose night carries Purnima is chosen, and the Moon's rising is the moment that matters. The Kojagari vigil of Bengal and Mithila is kept for Lakshmi, who is said to ask 'who is awake' (ko jagarti); in Braj the night recalls the raas of Krishna. Kheer is left in the moonlight and eaten the next morning. When Purnima begins in the day and ends the next morning, as it often does, the night belongs to the first day, while the day on which Purnima is running at sunrise, listed as Ashwina Purnima, is the day for the Purnima vrat and for bathing and gifts (snana-dana): one full moon, kept on two days.
How it is observed
A night vigil with Lakshmi worship, and kheer kept under the full moon.