What significance does religion put on fasting? How important is it?

The Christian viewpoint - short version:

The interior penance of the Christian can be expressed in many and various ways. Scripture and the Fathers insist above all on three forms, fasting, prayer, and almsgiving.
From the Catechism of the Catholic Church.

Then there is Ramadan: http://en.wikipedia.org/wiki/Ramadan

Thanks Needkarma.