Significance of Amavasya

Amavasya, or Amavasi, is the name of new moon night in Hindu religion. It is the first night of the first quarter of the lunar month. Since the moon is invisible on the day, Amavasya is also referred as no moon night. Amavasi holds great importance in Hinduism. Many Hindus choose this day to make offering (tharpanam) to the dead ancestors. There is also an ongoing debate whether Amavasya is auspicious or inauspicious.

The term ‘Amavasya’ is commonly used in all regional languages in India. The fortnight that starts with Amavasya is also referred as the Shukla paksha (bright half of the month). Mauni Amavasya in Hindu Magh month (January – February) and Mahalya Amvasya in Ashwayuja (September – October) are highly auspicious. In some regions people observe partial fast (Upvaas) or a complete fast on the day.

Thanks to mainstream Indian movies, Amavasya is considered the ideal day to perform black magic and evil acts. Since there is no moon, Amavasya night is pitch dark and this provides the ideal setting to invoke evil powers.
Earlier it was advised not to travel on an Amavasya night. The logic being that there is no moonlight and this could invite lot of hardship and danger.
Many astrologers advise against performing any important ceremony on the day as moon and important planet in Hindu astrology is not visible. In some regions, people do perform certain auspicious deeds on the day. But largely no new beginnings or important ceremonies are held on the day.

Symbolically, the period from Amavasya to Purnima (full moon) is considered to be the gradual awakening and transcendence into the fullness. From darkness to the gradual realization of the Supreme Soul.
From: http://www.hindu-blog.com/2008/02/significance-of-amavasya.html