Shani Ashtami, or Sani Astami, is observed on Ashtami day which falls on Shanivar or Saturday in a Hindu month. It may come either in Shukla Paksha or Krishna Paksha or in both. Shani Ashtami vrat is dedicated to Lord Shani who is known as the crucial celestial Lord in Hinduism and Hindu Astrology.
Devotees observed Shani Puja, Shani Japam, Shani Shanti homam, or Shani Thailabhishekam on Shani Ashtami to please Lord Sani.
In Andhra Pradesh and Karnataka, Venkateshwara swamy (Tirumala Tirupati Balaji) is also worshipped on Shanivar Ashtami. Venkateswara Vratam is also observed on this day. They believe that Venkateswara Puja can get peace to Shani and thus happiness in their life.
Some communities also worship Goddess Durga Bhavani on Shani Ashtami. Ashtami is dedicated to Goddess Durga and hence Durga Puja on Shani Ashtami can also pacify devotee’s Shani dosham.
Shani Ashtami vrat dates in 2012 - Link