Why did Avraham cut the animals into two?
Rashi: Because he was entering into a covenant with HaSh-m (that his children would inherit Eretz Yisrael), 1 and it was the way of people who entered into a covenant to cut up animals 2 and to pass between the pieces. 3
Ramban: This Pasuk also hints at all the animals that would be eligible to be brought as Korbanos, and every animal Korban was cut into pieces. 4 The Olah was dissected and entirely burned; the chest, right thigh and Chalavim of Shelamim must be separated, and Chalavim of Chatas and Asham must be separated.
Maharal (Gevuros Hashem Ch. 22, p. 96): This covenant was made with Midas ha'Din, so that any future sin would not alter it. Cutting (Gezeirah) of the animals alludes to Midas ha'Din
See verse 15:18.
Presumably that is why they are called "Korsei Bris."
Gur Aryeh: But perhaps the command to take animals was solely to teach Avraham about the Korbanos? If so, HaSh-m should have taught him orally about the Korbanos, without demonstrating. Since Avraham actually split the animals and passed between the parts, we learn that the main purpose of this event was to establish a covenant with HaSh-m. The particular types of animals alluded to the Korbanos.
Maharal (Gevuros Hashem Ch. 8, p. 47) writes similarly.
Why did Avraham not cut the birds into two?
Rashi: Because the nations of the world are compared to bulls, rams and goats, 1 and this is a hint that they will gradually be destroyed (like the bulls on Sukos, which decrease in number); whereas Yisrael are compared to doves. 2
Ramban: Bird offerings are not cut into pieces. 3
Rosh: Since there were two birds, he could put half on each side without cutting either. There were three of each kind of animal, so he needed to cut one of each.
One opinion in the Midrash explains that Avraham did not divide the birds because they represented the Chatas ha'Of. In the Melikah of the Chatas ha'Of, the Kohen would not sever the head of the bird [by cutting both Simanim] (Vayikra 5:8). What is a reason behind this detail in the Halachah?
Maharal (Gevuros Hashem Ch. 8, p. 47): Only a significant item can be split, and a bird is less significant than an animal. (An Olas ha'Of, however, would be split, because of the significance of an Olah, which is burnt completely on the Mizbe'ach.)
What do the animal carcasses and birds represent, and why is this alluded to at this point?
Maharal (Gevuros Hashem Ch. 8, p. 49):This important event can be interpreted in a number of ways. This was the beginning of HaSh-m's covenant with Avraham, and it alludes to all future events. Anything that has a solid beginning retains hope to return to its original lofty state. At times that Am Yisrael might be as if carcasses without bones and tendons, i.e. without merits of their own to hold them up, the merit of Avraham will stand in their favor and help them to do Teshuvah. Am Yisrael are compared to birds, for they humbly belittle themselves.
Why does this verse say "Tzipor" (in the singular)? There were two birds!
Ibn Ezra: "Tzipor" can refer to a number of birds.
Da'as Zekenim (15:12), Yeshu'os Meshicho: There was only one bird. "Tor" is Aramaic for "bull;" it symbolizes Bavel.
QUESTIONS ON RASHI
Rashi writes: "... The nations are compared to bulls, rams and goats... and Yisrael is compared to a dove...." Why is this so?
Gur Aryeh #1: The nations of the world are larger in size, like the animals, while Am Yisrael is small.
Gur Aryeh #2: The animals have a thick, dense body, so too, the nations are heavy with physicality. The bird is lightweight and able to fly, so too is Am Yisrael more spiritual.