How is the rainbow a sign of peace (that HaSh-m would never destroy the world)?
Ramban: (a) The rainbow always points away (to shoot arrows away from the world); and (b) it has no string.
Riva: It is made of fire and water (which normally cannot exist together), and HaSh-m made peace between them.
Why does the Torah write the word "Nasati" in the past tense?
Ramban 1 : Although it would seem that the rainbow was a new creation, in fact HaSh-m placed it in the clouds 2 from the time of the Creation of the world. 3 Now, however, was the first time that it was actually seen. What the Pasuk is therefore saying is, "The rainbow which I placed in the clouds at the time of the Creation, will from now on serve as a covenant between Me and you, so that whenever I see it, I will be remind Myself of the covenant of peace that is between us."
A similar idea is conveyed by the Seforno.
To conform to the Greek theory, that the rainbow is a natural phenomenon that is caused by the sun working in conjunction with the wet air. And this also explains why one can sometimes see a rainbow if one places a glass of water in the sun (Ramban).
Indeed, the Mishnah in Pirkei Avos (5:9) includes it in the list of ten things that were created Erev Shabbos at Bein-ha'Shemashos.
What exactly was the sign? It would seem that the rainbow is a natural occurrence due to the refraction of the sun's rays. How, then, does it serve as a sign of the covenant?
Rashi and Ramban: The rainbow was the visible sign of the covenant. 1
Seforno (to 9:13): The sign was the unnatural phenomenon of the fact that, when a double rainbow appears, the colors of the second rainbow appear in the reverse order than those of the first one. 2
Gur Aryeh: Before the Flood, the sun's glare was not as intense; this contributed to the Flood. After the Flood, that which had been blocking the glare was removed, and the intensified light was now capable of producing a rainbow; this was a sign of the covenant.
Maharal (Chidushei Agados Vol. 1 p. 127, to Rosh Hashanah 23b): Although the rainbow is natural, it does not occur every day, rather only when various factors are arranged properly. This specific arrangement is the sign of the covenant. The rainbow is used to describe the vision of HaSh-m's glory (Yechezkel 1:28). When it is visible in the world, it serves as a sign that HaSh-m will not destroy the world, for the sake of His glory therein. 3
What is the significance of a sign?
Ramban: Anything that is placed between two people to remind them of an agreement that they arrived at, is called a sign. 1
Seforno: The sign (in this case) is to remind the Tzadikim of that generation that the people are guilty (and worthy of destruction), to arouse them to pray and admonish the sinners.
Why is the word "l'Doros Olam" lacking a letter 'Vov'?
Rashi: This is a hint that some generations will not require a sign - such as those of Chizkiyahu ha'Melech and R. Shimon bar Yochai, since they were complete Tzadikim 1 (not worthy of destruction).
Maharal (Chidushei Agados Vol. 1, p. 159, to Kesuvos 77b): Most creations do not remain in existence on their own merit, rather, they receive HaSh-m's constant protection from harm, for He does not wish to destroy the world. This is represented by the rainbow. These righteous individuals had sufficient merit to justify their own existence.
Why is the word "l'Doros Olam" lacking a 'Vov'?
Rashi: This is a hint that some generations will not require a sign - such as those of Chizkiyahu ha'Melech and R. Shimon bar Yochai, since they were complete Tzadikim (not worthy of destruction).
רש"י: שיש דורות שלא הוצרכו לאות: קשה שהקשת נעשית בטבע מניצוץ השמש?
גור אריה: לפני המבול לא היה כח בלהט השמש לעשות קשת ולכן יכל המבול לבא, ולאחר המבול הוסר הדבר המונע את להט השמש ומזה נעשה קשת, וזהו אות הברית. 1
מהר"ל (חידושי אגדות ח"א, עמ' קכז ד"ה מעולם, ראש השנה בשלהי המסכת): הקשת טבעית, אבל צריכה להיות הכנה בבריאה לקשת וההכנה היא אות הברית, כי הקשת היא מראה כבוד ה', ובזכות כבוד ה' הדבק בהם יש לבריאה קיום.
וע"ע בבאר הגולה באריכות שאמנם דברים יוצאים אל הפועל באופן טבעי, אבל הטבע הוא שלוחו של הקב"ה, והקב"ה פועל את הכל, עיי"ש בבאר הרביעי, עמ' סג ד"ה אמנם, ושם עמ' סו ד"ה ועתה לגבי הקשת. וע"ע במהר"ל בחידושי אגדות שם ש'קשת' נקראת בשמה כי היא מידת הדין הקשה, ולכן היא אות ברית, עיי"ש.
רש"י: דורו של חזקיהו וכו': למה דורות אלו לא נזקקו לקשת?
מהר"ל (חידושי אגדות ח"א עמ' קנט ד"ה ומה שאמר, כתובות דף עז:): לכל הברואים יש קיום מהקב"ה שנתן אות שהבריאה לא תיפסד בחטא האדם, אבל לרשב"י וכדומה היה קיום בזכות עצמם.