Shani (Saturn) Amavasya is New Moon day on a Saturday. All Sun Moon Conjunction falling on a Saturday are called Shani Amavasya. This day is spent in devotion to Lord Saturn by many.
In 2010, Shani Amavasya falls on February 13th, June 12th and November 6th. June 12th is a day of special significance as this is also Shani Jayanti, birthday of Lord Saturn.