How the date is fixed
Sharad Navratri begins falls on the first day of the bright half of the lunar month Ashwina (Shukla Pratipada). The date is the day on which that tithi is running at sunrise (the usual rule, called udaya tithi).
About
Sharad Navratri begins on Ashwina Shukla Pratipada, the day after Sarva Pitru Amavasya, in September or October. It runs for nine tithis to Maha Navami and is followed by Vijayadashami on the tenth. The first day is the civil day whose sunrise carries Pratipada, and the ghatasthapana is done in the forenoon, or in Abhijit muhurta if the forenoon is unsuitable, avoiding Chitra nakshatra and Vaidhriti yoga. In Bengal the same days are Durga Puja, with the main rites from Shashthi to Dashami; in Gujarat they are the nights of garba.
How it is observed
Ghatasthapana on day one, worship of the nine forms of Durga, nine days of fasting, and kanya puja on Ashtami or Navami.