Here it is in verse 9-11
"I now establish my covenant with you and with your descendants after you 10 and with every living creature that was with you—the birds, the livestock and all the wild animals, all those that came out of the ark with you—every living creature on earth. 11 I establish my covenant with you: Never again will all life be cut off by the waters of a flood; never again will there be a flood to destroy the earth."
God always uses something as a symbol or sign of a convenant He is making. In the covenant He made with the Israelites, it was circumcism. In the new covenant He made with all who believe in His son, Jesus, it is water Baptism. In this covenant, it was a rainbow.
What I found interesting was in verse 14-15: "Whenever I bring clouds over the earth and the rainbow appears in the clouds, I will remember my covenant between me and you and all living creatures of every kind."
I always thought that the rainbow was for us to remember God's promise, but in truth, God says it's so He will remember. Then I wondered why God would need to remember anything. He's God after all. Then it dawned on me. God knew that in the distant future, He would once again look upon the earth and see nothing but violence and evil and be sorry once again that He made man. When in his anguish He would desire to wipe us out with a flood, He would pull from the heavenly files, the contract he made with Noah and fling it into the sky so that He wouldn't follow through with his desire.
So, the next time you see a rainbow, don't just stop and admire it's beauty, stop and consider that God put it there to stay His Almighty hand from wiping us out because I have a feeling things are far worse today than they were in Noah's Day.