Gemara

(Beraisa): 'Kidah' refers to bowing and putting one's face on the ground - "va'Tikod Bas Sheva Apayim Eretz";

'Kri'ah' is bending the knees - "mi'Kero'a Al Birkav";

'Hishtachava'ah' is prostrating on the hands and feet - "Lehishtachavos Lecha Artzah."

Berachos 12a (Rav): When one bows during Tefilah, he bends his knees at "Baruch" and straightens himself ba'Shem.

(Shmuel): He learns from "Hash-m Zokef Kefufim."

Question: "Mipnei Shmi Nichas Hu" (one should be submissive, i.e. bowed, when saying Hash-m's name!)

Answer: It does not say "bi'Shmi" (when saying My name), rather, "Mipnei Shmi" (before saying My name).

Rav Sheshes would bow like a stick. He would straighten himself slowly, like a snake.

28b (R. Tanchum): When praying, one must bow until all the vertebrae in the spine protrude;

(Ula): He must bow until he could see an Isar opposite his heart.

(R. Chanina): As long as he moves his head, this is not necessary.

(Rava): This is only if he is in pain and looks like he wants to bow.

34a (Beraisa): One bows in the beginning and end of Modim.

34b (Beraisa #1): It is praiseworthy to bow during Hoda'ah (Modim).

Contradiction (Beraisa #2): It is contemptible.

Resolution: It is praiseworthy to bow at the beginning of Modim, it is contemptible to bow at the end.

Rava, Rav Nachman and Rav Sheshes bowed at the beginning and end of Modim. They hold that 'Hoda'ah' in Beraisos #1 and #2 refers to Hallel.

Bava Kama 16a (Beraisa): One's spine becomes a snake after seven years if he does not bow at 'Modim'.

Rishonim

Rambam (Hilchos Tefilah 5:10): The four times one bows in Shemoneh Esre, he bows at Baruch and straightens himself ba'Shem.

Rambam (9:4): When the Shali'ach Tzibur reaches Modim, the Tzibur bows with him a little. They should not bow too much. They say 'Modim...'

Rosh (Berachos 1:15, from the Yerushalmi): Once, a Shali'ach Tzibur bowed too much, and R. Yochanan deposed him. Some say that he rebuked him.

Rosh (5:22): Rav Hai Gaon explains that one bows until he could see a coin on the ground even with his heart. Rava explains that moving the head suffices only if he looks like he bows, but he cannot bow properly due to illness.

Hagahos Ashri (Perek 4, Halachah 21): However, it is better to exert himself and bow more. One should not bow so much until his head reaches the belt of his pants.

Beis Yosef (OC 113 DH Kasuv): The Yerushalmi says so about Modim d'Rabanan. Many Rishonim say so about all bowings in Shemoneh Esre.

R. Yonah (Berachos 24a DH Ad): Rashi explains that one must bow until someone else standing opposite him can see only an Isar even with his heart. Rav Hai Gaon gave another Perush, that he bows until his navel, which is opposite his heart, makes a cavity the size of an Isar.

Tosfos (Bava Kama 16b DH v'Hu): One should bow, and straighten himself slowly, like a snake. If he did not, Midah k'Neged Midah, his spine becomes a snake. Some say that there is a bone in the spine from which one is revived, and if it becomes a snake, he will not be revived. It is unreasonable that the punishment for this is so great. All Yisrael have a share in the world to come!

Kaf ha'Chayim (113:18): The Zohar explicitly says that one who does not bow will not be revived.

Poskim

Shulchan Aruch (OC 113:4): One who prays must bow until all the vertebrae in the spine protrude.

Kaf ha'Chayim (16): This is to fulfill "all my bones (will say 'Hashem, who is like You?')' (Pri Chodosh, citing the Yerushalmi).

Shulchan Aruch (ibid): He does not bow from the midlle (his hips) and leave his head straight. Rather, also his head he bows like a reed.

Beis Yosef (DH u'Mah she'Chosav u'Piresh): Rashi explains that one must bow until there are two folds of flesh, one above and one below, with an Isar between them.

Beis Yosef (DH u'Mah she'Chosav v'Rabeinu): The Rashba says that he does not divide his height into two and bend. Rather, he bows until he could see a coin on the ground even with his heart.

Magen Avraham (4): When saying 'Baruch', he bends the knees. When saying 'Atah', he bows until the vertebrae protrude.

Shulchan Aruch (5): He does not bow so much until his head reaches the belt of his pants. If he is old or sick and cannot bow until the vertebrae in the spine protrude, it suffices to lean (lower) the head, since it is evident that he wants to bow, but it pains him.

Taz (5): It is haughtiness to bow more than the Shi'ur.

Divrei Chachamim (70, citing ha'Gaon R. C.P. Scheinberg Shlita): It suffices to bow 45 degrees. The Isur is to bow 90 degrees.

Shulchan Aruch (6): One bows quickly, at once. He straightens himself slowly, first his head, and afterwards his body, lest it looks like a burden.

Shulchan Aruch (7): When one bows, he bends his knees at "Baruch" and straightens himself ba'Shem.

Mishnah Berurah (12): When one bows in Modim, he bows his head and body like a reed at once. He stands bowed until Hash-m's name, and then he straightens up.

Kaf ha'Chayim (21): The Ari Zal says that one bows the body at Baruch, and the head at Atah. (In Modim, and Modim d'Rabanan, he bows the body at Modim, and the head at she'Atah.) Similarly, at 'Hashem', he straightens up first the body, and then the head. Even a sick person must bow (slightly) in two stages.

Shulchan Aruch (127:1): When the Shali'ach Tzibur reaches Modim, the Tzibur says Modim d'Rabanan and bows with him. They should not bow too much. Some say that one must bow also at the end of Modim d'Rabanan. It is good to be concerned for this opinion.

Beis Yosef (DH Kosav): The Tur explains that R. Yochanan objected only to the Shali'ach Tzibur bowing too much. The Rambam holds that the same applies to the Tzibur. Perhaps he rules like the latter opinion, and holds that R. Yochanan rebuked one of the Tzibur who bowed too much.

Taz (1 and Mishnah Berurah 2): One bows like in Shemoneh Esre. If one bows less, the Shulchan Aruch should have given the Shi'ur! The Bach says that one should just move his head a little. This is not the custom.

Kaf ha'Chayim (2): Many say that one bows fully, like in Shemoneh Esre. The Bach says that it suffices to bend the head, but he does not forbid full bowing. Therefore, one should bow fully.

Rema: Some say all of Modim d'Rabanan in one bowing. This is the custom.

Mishnah Berurah (5 and Bi'ur Halachah DH v'Yesh): People follow the Rema, but the Maharshal holds that one straightens up a little at Hash-m, and bows again at the end. Chayei Adam holds that this is primary.

Kaf ha'Chayim (10): The Ari Zal holds that one bows only at the beginning.

Kaf ha'Chayim (113:17): Ya'aros Devash says that bowing shows that one (his body) will return to the dirt, and his Ru'ach will return to Hash-m. One who does not bow to show humility that he will be dirt, his spine becomes a snake, which eats dirt. Ru'ach Chayim explains (Tehilim 119:25) "Davka Nafshi le'Afar Chayeni ki'Dvarecha" (since I bowed, like You said that one must bow lest his spine become a snake, You will revive me in Techiyas ha'Mesim).