We read in Parashat Noach of the "berit" ("covenant") God establishes with Noach and his offspring, by which He promises never again to flood the earth. God chooses the rainbow as a sign of this covenant: "I have set My bow in the clouds, and it shall serve as a sign of the covenant between Me and the earth. When I bring clouds over the earth, and the bow appears in the clouds, I will remember My covenant between Me and you and every living creature among all flesh" (9:13-15).
Rav Shimshon Refael Hirsch, in his commentary, briefly lists several traditional explanations of the symbolic meaning of the rainbow, and why it specifically was chosen to signify this berit between God and the earth. One approach sees the rainbow as a reversed weapon: the bow's string, from which the arrows are sent, faces away from the earth, rather than towards it, symbolizing the turning away of God's anger. He turns His "weapon" away from the earth, so-to-speak, having decided against destroying it. Others explained that the arc shape symbolizes the union between heaven and earth, the bond between man and God established through this treaty. More famously, perhaps, the rainbow is often seen as representing the presence of light among the darkest and most ominous clouds, and thus symbolizes the presence of divine grace and compassion even when God decrees calamity upon the earth.
Rav Hirsch himself, however, prefers a different approach, focusing instead on the rainbow's constitution. A rainbow, of course, is a single ray of light refracted into its various components, which assume different colors. Rav Hirsch claims that this manifestation symbolizes the wide gamut of existence – both human and animal, all of whose components are joined together in this covenant. Indeed, God emphasizes to Noach that when He sees the rainbow, He will "remember the everlasting covenant between God and all living creatures, all flesh that is on earth" (9:16). This covenant makes no distinctions between different types of creatures, and even among human beings, it applies to all races and classes. This universality of the covenant is symbolized by the refraction of a single ray of light into numerous different colors.
Rav Gavriel Zev Margolis, in his "Torat Gavriel," suggests an entirely different explanation for the significance of the rainbow. In the famous first chapter of Sefer Yechezkel, the prophet Yechezkel describes the "ma'aseh merkava" – the vision he beheld of God's glory transported by a chariot. The final verses of this prophecy describe the actual appearance of God's glory: "There was radiance all about him [the figure of a person described one verse earlier]. Like the appearance of the bow which shines in the clouds on a day of rain, such was the appearance of the surrounding radiance" (Yechezkel 1:27-28). While obviously we cannot possibly claim to have a concrete understanding of what Yechezkel actually saw, it is clear that the rainbow is part of the visible appearance of God's glory. Rav Margolis suggests that for this reason God chose the rainbow – a form of God's visible manifestation – as the sign of His covenant. This interpretation has strong basis in the wording of these verses. God tells Noach, "I have set MY BOW in the clouds." The reference to the rainbow as "God's bow" can be understood more clearly if God speaks here of the image of His physical manifestation to the prophets.
Why would God use a feature of His physical manifestation as a symbol of His covenant?
The rainbow serves as a means of reassurance to mankind that God will never again destroy the earth. As the Netziv writes in his commentary to these verses, God here does not establish a new covenant with Noach, but rather reaffirms the covenant He established at the time of creation that the earth will never cease to exist. We might suggest, then, that He accomplishes this by giving mankind a glimpse of His presence. This quasi-prophetic vision of the rainbow represents God's ongoing involvement and interest in the world. He will never destroy the earth because He maintains a relationship with it, as evidenced by His "appearance" in the form of the rainbow. Understandably, then, the rainbow serves as the eternal sign of the Almighty's guarantee to continue preserving the earth and its inhabitants.