The rainbow will be in the cloud, and I will look on it to remember the everlasting covenant between God and every living creature of all flesh that is on the earth. This is the sign of the covenant which I have established between Me and all flesh that is on the earth. Genesis 9:16,17
Noah was the last man to see this world before the destruction of the Flood, and he was the first to view it afterwards. Some scientists debate over whether there were such things as rainbows because some evidence indicates a vastly different world after the Flood.
Whether the rainbow is actually new is not the issue; by it God tells us He will never again "destroy every living thing" (8:21). He has attached His promise to an element in creation and it reminds us to trust His unbreakable promise.
Man's sin and violence had so corrupted creation that God swore to destroy both man and his world (6:5-13). In that respect, there is still very little difference between the world before or after the Flood. The Flood, of course, did not get rid of sin, it only reduced the number of sinners. From his earliest moments of life, fallen man still selfishly desires his own perverse ways (8:21; Ps. 51:5).
Through God's mercy (6:8) faithful Noah, "a preacher of righteousness" (2 Pet. 2:5), and his family were spared by means of the ark (Heb. 11:7). Noah's rescue serves as an example of salvation by means of Baptism. Our old sinful nature is destroyed by water and the Word, and a new life mercifully sprouts in the pattern of Christ's resurrection (1 Pet. 3:20,21; Rom. 6:3,4).
The word for "rainbow" in Scripture is simply "bow" as in "bow and arrow". Some have seen in this an indication God would not again wage war on His rebellious creation; now He lets His warrior's bow hang peacefully at His side (Ps. 18:14; 46:8,9; 76:1-6). What is certain is that while God acts to turn us from sins, frequently allowing trials and sorrow to come upon us because of them, He nevertheless is merciful and remembers His promises (9:16; Is. 54:7-10).
The Flood was God's act of discipline (destruction) and deliverance (Noah's ark). This anticipated the life of Jesus in which discipline (the punishment in the suffering of the cross) and deliverance (atonement for sin and victory over death) are lived out for all of humanity. The benefits of life and salvation won by Jesus are passed on to us through His Gospel and its attachment to the Sacramental elements of Baptism and the Lord's Supper. Christ's promises, joined with elements in creation, move us to trust in Him and live.