After the Flood, God spoke to Noah and, much like Adam, made a covenant with Noah. He repeated the command to multiply and replenish the earth and established the principle of capital punishment. But more particularly, He made glorious promises.
In verse 8, we see something that we have seen before and consistently: God is the initiator of the covenant. Noah, like Adam, had the covenant imposed upon him. God always maintains the initiative in dealing with His creation. There is something different about this covenant though. God includes in the covenant not only Noah, but also his seed and even the animals. Even though the human race was almost wiped out, God promises that the descendants of Noah would be preserved. This is a glorious promise because in it God declares that the schemes of the enemy will not prevail - God promises to win! There is another difference with this covenant. In the covenant with Adam, the terms were stark: obey and live, disobey and die. In this covenant, God unilaterally promises to perform what He promises without further conditions. God would have been perfectly just to wipe out everyone and everything. Yet, in an act of amazing grace, He promises to preserve the race.
Another act of God's grace was to ordain a sign of the covenant: the rainbow. Like all of the signs and seals of God's covenants, the bow in the sky has a direct relationship between the sign and the thing signified. In this case, God sets down His battle-bow and no longer threatens the earth with the waters of judgment. No matter how violent the storm is, and no matter how dark the looming clouds are, God will not destroy the earth with water. The text also tells us that the purpose is so that God will see the bow and remember. What an act of condescending love! Of course, God does not - cannot - forget His covenant, but the rainbow in the clouds reminds us that He will not forget His promise.
So when you see the bow in the clouds after the storm, remember. Remember that God is a God of mercy, even in the midst of judgment. Remember that God's plan will not be frustrated by sin or the tempter. He will prevail. And remember that all of this is according to His sovereign decree and for His glory.