Gemara

33b - Mishnah: Just like the Shali'ach Tzibur is obligated (to say Shemoneh Esre), every individual is obligated;

R. Shimon ben Gamliel says, the Shali'ach Tzibur is Motzi the Rabim.

34b: If an individual did not bless (say Shemoneh Esre), another may not bless for him.

Beraisa - R. Gamliel: If the Shali'ach Tzibur is not Motzi the Rabim, why does he say Shemoneh Esre out loud?

Chachamim: He is Motzi one who is not Baki (does not know the Berachos).

R. Gamliel: Just like he is Motzi one who is not Baki, he is Motzi a Baki.

R. Yochanan: Chachamim (retracted and now) agree with R. Gamliel.

Rav: The argument persists.

Question: But R. Yochanan taught that the Halachah follows R. Gamliel. This implies that they argue!

Answer #1 (R. Aba): Chachamim agree about the Berachos of Rosh Hashanah and Yom Kipur (of Yovel; its Berachos are like those of Rosh Hashanah). They argue about the rest of the year.

Objection: R. Chanah said in the name of R. Yochanan that the Halachah follows R. Gamliel regarding the Berachos of Rosh Hashanah and Yom Kipur!

Answer #2 (Rav Nachman bar Yitzchak): R. Yochanan meant that R. Meir agrees with R. Gamliel. He needed to say that the Halachah follows R. Gamliel, because Chachamim argue:

Beraisa - R. Meir: The Shali'ach Tzibur is Motzi the Rabim in the Berachos of Rosh Hashanah and Yom Kipur;

Chachamim say, just like the Shali'ach Tzibur is obligated, every individual is obligated.

Question: Why are the Berachos of Rosh Hashanah and Yom Kipur different?

Suggestion: There are many verses (in Musaf).

Rejection: Rav taught that it suffices to say 'And it is written in Your Torah'!

Answer: There are many Berachos.

Rav Acha bar Avira citing R. Shimon Chasida: R. Gamliel exempts even people in the fields, all the more so people here (in the Beis ha'Keneses).

Objection: It is more reasonable to exempt people in the field. They are Anusim (they cannot come)!

Aba brei d'Rav Minyamin bar Chiya - Beraisa: People in back of the Kohanim are not blessed (even though Yisraelim in the field are blessed)!

Correction - Ravin citing R. Shimon Chasida: R. Gamliel exempts only people in the fields, who are Anusim due to Melachah.

Berachos 45b - Beraisa: If two ate together, each should bless by himself.

This is if both are Chachamim; if one is a Chacham and the other an ignoramus, the ignoramus should be Yotzei by listening to the Chacham.

Rishonim

Rif: We learn that the Halachah follows R. Gamliel only regarding the Berachos of Rosh Hashanah and Yom Kipur. The rest of the year, just like the Shali'ach Tzibur is obligated, every Baki is obligated. The Shali'ach Tzibur is also Motzi people in the fields who cannot come to the Beis ha'Keneses. People in the city are not Yotzei unless they come and hear the Shali'ach Tzibur from the beginning to the end. The Shali'ach Tzibur is Motzi people only in Rosh Hashanah and Yom Kipur because there are many Berachos, i.e. nine. A regular Shabbos or Yom Tov has only seven Berachos.

Ran (DH Yachid) and Hagahos Ashri (Sof Perek 3): Regarding all Berachos, even one who was Yotzei can be Motzi. The Yerushalmi says that there are three exceptions: Birkas ha'Mazon, Kri'as Shma and Tefilah. It is fitting for everyone to say Shemoneh Esre himself, for it is a request for mercy. However, if one is not Baki, the Gemara (Berachos 45b) says that a Baki is Motzi one who is not Baki.

Question (Ran Megilah 13b DH Aval): Ge'onim say that a Minyan is required for Birkos Kri'as Shma because it contains Kedushah. This implies that with a Minyan, a Shali'ach Tzibur can be Motzi even a Baki in Shemoneh Esre. The end of Rosh Hashanah says otherwise!

Answer (and Rebuttal of Ran - Tur OC 594 citing Ritz Gei'us and Rav Shrirah Gaon): A Baki is Motzi one who is not Baki in Berachos over Mitzvos and Hana'ah, which are normally said aloud. Shemoneh Esre was enacted to be whispered (so no one else can hear it), so only a Shali'ach Tzibur may be Motzi another.

Ran (DH Masnisin): R. Gamliel and Chachamim argue about all Berachos and Tefilos, of Rosh Hashanah and the entire year.

Ran (DH Gemara): Since it is necessary for the Shali'ach Tzibur to say Shemoneh Esre out loud to be Motzi people who are not Baki, it is like an obligation, therefore he is Motzi even a Baki.

Rambam (Hilchos Berachos 8:9): A Shali'ach Tzibur is Motzi the Rabim. If he prays and they answer Amen to every Berachah, it is as if they pray. This refers to one who is not Baki. A Baki is Yotzei only through his own prayer.

Rambam (10): This refers to the rest of the year except for Rosh Hashanah and Yom Kipur of Yovel. At these two times a Shali'ach Tzibur is Motzi a Baki just like he is Motzi one who is not Baki. This is because the Berachos are long and most people cannot concentrate like a Shali'ach Tzibur. Therefore, one may rely on the Shali'ach Tzibur.

Question: The Gemara said that the Shali'ach Tzibur is Motzi only people in the fields who cannot come. Why did the Rambam omit this?

Answer (Kesef Mishneh): The Rambam explains that R. Gamliel exempts only people in the fields during the rest of the year. He cannot refer to Rosh Hashanah and Yom Kipur, for we said that they are Anusim due to Melachah! Since the Halachah follows Chachamim the rest of the year, there is no distinction between people in the fields and in the city.

Rosh (Rosh Hashanah 4:14): R. Gamliel says that a Shali'ach Tzibur is Motzi even a Baki. BaHaG and the She'altos (va'Yetzei) learn from here that if omitted mention of Rosh Chodesh or made any mistake which requires him to repeat Shemoneh Esre, he should be Yotzei by listening to the Shali'ach Tzibur's repetition from the beginning to the end. Even though R. Gamliel exempts only people in the fields who could not come, but not people in the city, that refers to people who did not pray at all. The Halachah follows R. Gamliel only on Rosh Hashanah and Yom Kipur, for there are many Berachos. This refers to Musaf. At all other times the Halachah follows Chachamim, and a Shali'ach Tzibur is not Motzi a Baki.

Poskim

Shulchan Aruch (OC 124:1): After the Tzibur finishes Shemoneh Esre, the Shali'ach Tzibur repeats it. This is in case there is someone who does not know how to pray; such a person listens and is Yotzei. He must hearken to everything the Shali'ach Tzibur says from beginning to end. He does not interrupt or talk. He takes three steps back at the end like one who prayed himself.

Beis Yosef (DH ul'Acher): The one listening must understand what is said. If not, he was not Yotzei.

Beis Yosef (69 DH Tanan, citing R. Yonah, ibid.): Regarding Shemoneh Esre, the Shali'ach Tzibur is Motzi people in the fields who are unable to come to the Beis ha'Keneses.

Chazon Ish (19:3 DH veha'Einam): Even Chachamim agree that people in the fields are Yotzei. It seems that people who do not understand Lashon ha'Kodesh, who are not Yotzei through listening are also Anusim. They should be Yotzei just like people in the fields! However, the Beis Yosef connotes otherwise.

Mishnah Berurah (2): One who says Shemoneh Esre himself is Yotzei even if he does not understand. However, he should understand at least the first Berachah.

Beis Yosef (DH v'Chosav Rabeinu): R. Yerucham says that one is Yotzei from a Shali'ach Tzibur only with a Minyan. An individual cannot be Motzi an individual. It is fitting for everyone to say Shemoneh Esre himself, for it is a request for mercy. However, a Baki can be Motzi one who is not Baki. This is unlike the Ritz Gei'us, who says that the Heter for a Baki to be Motzi an ignoramus applies only to Berachos of Hana'ah.

Shulchan Aruch (10): If one made any mistake which requires him to repeat Shemoneh Esre, he should be Yotzei by listening to every word of the repetition as if he was praying himself.... Since he prayed but erred, the Shali'ach Tzibur is Motzi him even though he is Baki.

Question: R. Gamliel (Rosh Hashanah 34b) says that a Shali'ach Tzibur is Motzi even a Baki, but the Halachah follows Chachamim, except for Rosh Hashanah and Yom Kipur!

Answer #1 (Beis Yosef DH v'Tamhani): Perhaps Chachamim argue only about someone who did not pray.

Question (Taz 5): Chachamim say that people in the fields are not Yotzei, even though they are Anusim. Surely one who erred b'Shogeg is no better!

Answer #1 (Taz ibid.): Chachamim say that a Baki may not be Yotzei all of Shemoneh Esre through the Shali'ach Tzibur, but they agree that one who prayed may be Yotzei one matter that he erred about.

Answer #2 (Taz ibid.): All agree that one who hears the repetition is Yotzei. They argue only about people who do not hear it.

Sha'arei Teshuvah (114:3): If one erred regarding Meshiv ha'Ru'ach u'Morid ha'Geshem, he was not Yotzei the first three Berachos (they are like one), so it is as if he did not pray at all. If so, perhaps he cannot be Yotzei through the Shali'ach Tzibur.

Mishnah Berurah (124:39): The Shulchan Aruch connotes that one is Yotzei through the repetition no matter what mistake he made.

Mishnah Berurah (40,41): Nowadays it is better to pray again, for not everyone can hear every word from the Shali'ach Tzibur; perhaps he will miss a word that is Me'akev. If the Shali'ach Tzibur normally swallows words, or does not say all of Modim out loud, one must pray again himself.

Kaf ha'Chayim (56): One may rely on being Yotzei through the Shali'ach Tzibur only if one knows that the Shali'ach Tzibur has Yir'as Shomayim and says every letter of the repetition audibly and says each word slowly by itself. Also, the listener must be able to listen without any stray thoughts.

Kaf ha'Chayim (54): One who is unsure whether or not he prayed may not be Shali'ach Tzibur, for the Shali'ach Tzibur must (Vadai) pray silently first. However, if one prayed Shemoneh Esre but must repeat it because of a mistake, this is considered that he prayed silently.

Shulchan Aruch (594:1): If Reuven did not pray nine Berachos (of Musaf of Rosh Hashanah), Shimon cannot be Motzi him (without a Minyan).

Mishnah Berurah (2): This is even if Shimon needs to pray now and Reuven is not Baki.

Kaf ha'Chayim (4): Some say that Shimon may say the words out loud and Reuven may repeat them. On Rosh Hashanah and Yom Kipur the custom is to pray audibly. Normally, it is better not to pray audibly. One may be lenient to enable the other to pray.

Rema: Some say that if Reuven is not Baki, Shimon can be Motzi him.