Gemara

50a (Mishnah): In a place where the custom is to work on Erev Pesach until midday, one does so.

50b - Question: Why does the Mishnah discuss Erev Pesach? The same applies to any every Erev Yom Tov or Erev Shabbos!

(Beraisa): If one works after [the time for] Minchah on Erev Shabbos or Erev Yom Tov, his work will not be blessed.

Answer: Any other Erev Yom Tov or Erev Shabbos is forbidden only after Minchah. Erev Pesach is forbidden after midday.

58a (Mishnah): The [afternoon] Tamid is slaughtered at eight and a half hours of the day and it is [totally] offered at nine and a half hours;

When Erev Pesach is on Friday, it is slaughtered at six and a half hours.

Question: What is the source of the first law?

Answer #2 (Rava): Mid'Oraisa, the Tamid may be offered from when shadows point east;

We learn from "Bein ha'Arbayim", i.e. from when the sun is in Ma'arav (the west);

We delay the Tamid two hours until eight and a half hours, to allow more time to offer voluntary Korbanos, which cannot be offered after the Tamid.

Berachos 26b (Beraisa): Minchah Gedolah (the earliest time for Minchah) begins at six and a half hours of the day.

Minchah Katanah begins at nine and a half hours of the day.

Rishonim

Teshuvas ha'Rif (320): Tefilas Minchah is from six hours, for it corresponds to the afternoon Tamid. It is offered when shadows point east. So it says in the Mishnah, that Minchah is from six and a half hours. One who did not pray then, he prays until evening.

Rambam (Hilchos Tefilah 3:2): The time for Tefilas Minchah corresponds to the afternoon Tamid. Every day it was offered at nine and a half hours. This is called Minchah Ketanah. When Erev Pesach falls on Erev Shabbos, they would slaughter the Tamid at six and a half hours. Therefore, one who prayed after this time was Yotzei. From this time the Chiyuv came. This is called Minchah Gedolah.

Teshuvas ha'Rosh (4:9): Tefilas Minchah is from six and a half hours. If one prays Minchah Ketanah and Minchah Gedolah, he is Yotzei his Chiyuv through the first. We say (Berachos 28b) that if the time for Minchah came before one prayed Musaf, he prays Minchah first.

Rashi (Shemos 12:6 DH Bein): Bein ha'Arbayim (the time to slaughter Pesach) is after six hours, when the sun leans to the west.

Pri Chodosh (Likutim, OC 231): Many think that letter of the law, Tefilas Minchah is from six and a half hours, like it says in Berachos. Rashi on the Gemara connotes like this. Really, the primary time starts after midday, when shadows start pointing east. Rashi on Chumash says so. Pesachim 50b connotes like this. It asked why the Mishnah discusses Erev Pesach, since every Erev Yom Tov or Erev Shabbos one may not do Melachah from Minchah. If this is only from six and a half hours, what was the question? On Erev Pesach, we forbid after midday! Do not say that this was the Gemara's answer. The answer was too obvious!

Note: Perhaps the questioner thought that we forbid within a half-hour of Minchah on every Erev Shabbos and Yom Tov, just the Mishnah did not bother to say so. Tosfos (Pesachim 99b DH Min) says like this.

Pri Chodosh: Rather, Zman Minchah is after midday. The Gemara answered that Erev Shabbos and Erev Yom Tov are forbidden only from Minchah Ketanah, like Rashi says. In Berachos 21a, we say that one may pray the entire day. If Minchah begins at six and a half hours, there is a half hour after midday when one may not pray! Yoma 28b says that the primary time is from midday, just we wait a half hour until it is evident that the shadows lean [to the east]. If one prayed in the half hour after midday, he was Yotzei. Tosfos (58a DH Mukminan) says so. This is unlike the Beis Yosef, Bach and Tur. The Rambam (Hilchos Tefilah 3:9) says that if one omitted Shacharis, and midday passed, he prays Minchah. The Kesef Mishneh says that he was imprecise. We need not say so. One could refute all my proofs. Just like Chachamim decreed not to slaughter the Tamid before six and a half hours, the same applies to Minchah.

Machazik Berachah (233:1, and Kuntres Acharon): The Yerushalmi supports the Pri Chodosh. Also Zichron Yosef and Teshuvas ha'Rif says so.

Note: I do not understand the proof from Teshuvas ha'Rif.

Poskim

Shulchan Aruch (OC 233:1): If one prayed Minchah after six and a half hours, he was Yotzei.

Beis Yosef (DH u'Zmanah): Rashi explains that one may not offer the afternoon Tamid earlier than six and a half hours, for it says Bein ha'Arbayim. This is from when afternoon shadows lean, from when the sun leans to the west, at six and a half hours.

Taz (1): Here, the Mechaber wrote Stam like the Rambam. Our custom is that often we pray Minchah before Plag ha'Minchah, i.e. 10 and a half hours.

Hagahah (in Shulchan Aruch Hadras Kodesh, 1): Toras Chayim emends the text to say "Minchah Gedolah, i.e. six and a half hours. I.e. we do so on Shabbos that is Erev Yom Tov [to enable eating Seudah Shelishis after praying Minchah, but before Minchah Ketanah, in order to enter Yom Tov with appetite - PF], or when one will have a large meal."

Magen Avraham (1): This connotes that before this, he was not Yotzei. Even though the time is really from six hours, like it says in Yoma 28b, since we are not experts, he was not Yotzei.

Gra (DH Mi): We learn from Berachos 26b. The Rambam says that Minchah Gedolah is b'Di'eved. Rashi holds that it is l'Chatchilah, but the primary time is Minchah Ketanah. The Tur and other Poskim hold like this.

Mishnah Berurah (1): The primary time is from nine and a half hours. If one prayed from six and a half hours, he was Yotzei. Some Rishonim are lenient l'Chatchilah from then. If one wants to eat or travel, and now he can pray with the Tzibur, but he will not be able to after Minchah Ketanah, all permit Minchah Gedolah l'Chatchilah.

Mishnah Berurah (2): This connotes that before six and a half hours, even b'Di'eved he was not Yotzei. The Magen Avraham, Derech ha'Chayim and Birkei Yosef say so. Some Acharonim say that l'Chatchilah one may not pray in the half-hour after midday, but if he did, b'Di'eved he does not pray again.

Kaf ha'Chayim (1): Some are lenient to pray from six hours when there is a need. One should not pray before nine and a half hours without a need, like the Rambam and Mechaber.

Shulchan Aruch (ibid.): The primary time is from nine and a half hours.

Beis Yosef (DH umi'Divrei): The Tur connotes that l'Chatchilah Minchah is from six and a half hours. Also Teshuvas ha'Rosh connotes like this. However, the Rambam holds that the primary time is from nine and a half hours. Minchah Gedolah is only b'Di'eved. The Tur in Siman 234 connotes like this. He says that it is proper to pray Reshus (an optional Tefilah) only Minchah Gedolah, for it corresponds to something that is not constant every day.

Magen Avraham (2): The primary time of Shechitah mid'Oraisa is from six and a half hours, just normally they used to offer it at nine and a half hours, in order to offer beforehand Korbanos Yachid, which are forbidden afterwards. (If one wants to eat, l'Chatchilah he prays Minchah Gedolah, like I wrote (232:16), and all the more so one who wants to travel (89:8).)

Magen Avraham (Reish Siman 232): It is called Minchah because the ideal time is when the Minchah [that accompanies the Tamid] was offered. This supports that nine and a half hours was the primary time. Sometimes Shechitah was at six and a half hours, but the Minchah was never offered then!

Kaf ha'Chayim (3): It is better to pray from Plag ha'Minchah (10 and three quarter hours) than from the beginning of Minchah Ketanah.

Rema: We calculate these hours according to the length of the day.

Magen Avraham (3): Terumas ha'Deshen (1) connotes that we calculate [hours] from dawn until Tzeis ha'Kochavim, like I wrote (443:3, regarding Chametz). The Levush, Lechem Chamudos and Shiltei ha'Giborim (on the Mordechai Berachos Perek 4, 3) calculate from sunrise to sunset. Pesachim 93b supports them, for Shechitah is Kosher only until the beginning of Shki'ah (Tosfos Menachos 20b DH Nifsal), which is an hour and a fifth before Tzeis ha'Kochavim. However, perhaps they enacted based on when they offered the Minchah, like I wrote (232). Also the Levush refers to the end of Shki'ah, which is about a quarter hour before Tzeis ha'Kochavim.

Gra (DH u'Mesha'arinan): Pesachim 4b forbids Chametz after six hours. These must be relative hours, like it says there (we learn from the Isur to slaughter Pesach while one has Chametz). We say (12b) that in the seventh hour, the sun is in the west. Six hours it is in the middle of the sky. Avodah Zarah 25a [calculates how much was added to the 12 hours of daylight when the sun stood still for Yehoshua, even though it was in summer].

Mishnah Berurah (3): In this Siman and Siman 443, the Mechaber wrote Stam that we calculate the hours until Tzeis ha'Kochavim, so the time for Minchah is until Tzeis ha'Kochavim according to Rabanan, and until one and a quarter hours before this according to R. Yehudah. Some calculate from sunrise to sunset.

Kaf ha'Chayim (7): For Tefilah, the custom is to be lenient and count the hours from sunrise to sunset.