The rainbow is the sign of God's promise.

Noah was a righteous man whom God told to build a boat or "ark" to save his family and every kind of animal, bird, and creeping thing from the flood that would destroy life on the earth.  Noah did so. It rained for 40 days and nights. When the water receded, the ark landed on Mount Ararat. Noah sent out a dove to see if the flood waters had subsided. At first the dove returned, then the dove brought an olive branch, and finally the dove did not return. After this God put the rainbow in the sky as a covenant promise that he would not destroy the earth again by a flood.

Gen 6:9-22 - Command to build ark
Gen 9:1-17 - Rainbow promise