Gemara

(Mishnah - Chachamim): In Yehudah, people would do Melachah on Erev Pesach until midday. In Galil, they would not do Melachah at all.

Beis Shamai forbid the night before;

Beis Hillel permit until sunrise.

(Gemara) Question: The Tana first says that it is only a custom [to refrain from work], and then he discusses when it is forbidden!

Answer (R. Yochanan): The first Tana is R. Meir, and the latter Tana is R. Yehudah:

(Beraisa - R. Yehudah): In Yehudah, people used to do Melachah on Erev Pesach until midday. In Galil, they would not do at all.

R. Meir: It does not depend on Yehudah and Galil. Rather, where the custom is to work, one may work. Where the custom is not to work, one may not work.

Inference: R. Meir says that it was a custom. This implies that R. Yehudah holds that it is forbidden!

(Mishnah - R. Meir): If one started any Melachah before the 14th he may finish it on the 14th;

One may not start a Melachah on the 14th even if he could finish it the same day.

Chachamim permit three professions to work on the 14th until midday - tailors, barbers and launderers.

55b (Beraisa - R. Meir): On the 14th one may finish any Melachah for the sake of the Mo'ed (festival). If it is not for the Mo'ed it is forbidden;

Melachah is permitted on Erev Pesach until midday where this is the custom.

Inference: Where this is not the custom, it is forbidden even for the sake of the Mo'ed!

Conclusion: The Mishnah permits only for the Mo'ed.

(Mishnah - Chachamim): Three professions...

(Beraisa): Tailors are permitted because commoners may sew normally on Chol ha'Mo'ed. (Erev Pesach is more lenient than Chol ha'Mo'ed);

Barbers and launderers are permitted, for one who came from overseas or was freed from prison may take haircuts and launder on Chol ha'Mo'ed;

The Tana does not learn starting a Melachah from finishing it. (He permits only things that one may start during Chol ha'Mo'ed.)

Rishonim

Rambam (Hilchos Yom Tov 8:19): Even where the custom is to do Melachah [on Erev Pesach], one may not begin a Melachah on the 14th, even if he can finish it before midday. Only three professionals may begin where the custom is to work, and they may work until midday - tailors, barbers and launderers. Other professionals, if they already started before the 14th, they may finish until midday. People do not need other professionals so much.

Rebuttal (Ra'avad): R. Meir holds that any Melachah needed for the Mo'ed, one may finish it on the 14th, even where the custom is not to work. What is not needed for the Mo'ed is permitted where the custom is to work. Rabanan say that these three professions may even begin, even where the custom is not to work. They come to permit, and not to forbid, since they did not say "only three professions may work..." The Tosefta (3:14) clearly says so. The Rif says so. This is primary.

Rosh (4:5): R. Meir holds that Beis Hillel and Beis Shamai do not argue about this, for it is only based on custom. There cannot be an argument about a custom. We just see what people sdo! Even though the Halachah [normally] follows R. Yehudah against R. Meir, here the Halachah follows R. Meir. The Stam Mishnah at the beginning of the Perek is greater than Chachamim's opinion.

Rosh (ibid.): It seems that R. Meir permits finishing where the custom is not to work, i.e. only until midday. The ultimate need of the Mo'ed is cutting hair and laundering, therefore one may even begin them, but even so we permit them only until midday. R. Yonah permits the entire day, for the Seifa permits until midday. This implies that the Reisha permits the entire day for needs of the Mo'ed. This is no proof. It could not have taught any other way. Since it says "we may do Melachah on Erev Pesach", it needed to say "until midday."

Poskim

Shulchan Aruch (OC 468:3): Before midday, where the custom is to do Melachah on Erev Pesach, we do. Where the custom is not to do, we do not do.

Magen Avraham (4): The Re'em asked "if they err and think that it is forbidden, one may permit them! If they know that it is forbidden, but are stringent, any day of the year it is forbidden, for this is like a vow (YD 214:1)! (If so, why was this taught only regarding Erev Pesach? - PF) We must say that they know that it is permitted. The Shulchan Aruch's law affects one who came from elsewhere. He must obey the stringencies of his new place." I say that if a custom was for the sake of a fence, like here, their children may not change it. If they had a custom at another time of the year not to do Melachah, their children could change it. See Tosfos (51a).

Mishnah Berurah (10): It is permitted even for pay.

Mishnah Berurah (11): The entire night is permitted. The Isur begins at sunrise. Also their children for all generations may not work, due to "Al Titosh Toras Imecha." They cannot permit this [like a vow] through She'elah.

Kaf ha'Chayim (35): The Rashba (98) says that they cannot permit through She'elah.

Kaf ha'Chayim (30): Even where the custom is not to work, buying and selling is permitted until midday.

Kaf ha'Chayim (31): It is logical to be stringent in Yerushalayim, for some of those who gather there are from places where they do not work (Tosfos 14a DH Shtei).

Kaf ha'Chayim (32): Aderes Eliyahu Yisrael (9a) says that they are lenient to cut hair through a Yisrael in Yerushalayim, even though they are stringent about Melachah. We do not protest, for it is needed for the Mo'ed, like the Ra'avad, unlike the Shulchan Aruch. The same applies in Salonika. This is based on the Rema in Sa'if 5.

Kaf ha'Chayim (33): The Shulchan Aruch rules like the Rambam, who permits only needs of the Mo'ed. The Rema brings an opinion that permits even what is not needed for the Mo'ed.

Beis Yosef (DH veha'Roke'ach): The Roke'ach permits Melachah, even in one's house, through a Nochri. "Telling a Nochri is Shevus (forbidden mid'Rabanan)" applies only to Shabbos, Yom Tov and Chol ha'Mo'ed.

Rema: In our lands we do not do. This is only for absolute Melachos. Regarding other Melachos, there is not even a custom.

Taz (3) and Magen Avraham (6): The Levush says that in many places, the custom is to do even absolute Melachah.

Mishnah Berurah (12): Every place depends on its custom.

Shulchan Aruch (5): Even where the custom is to work, only three professions may even begin on the 14th. Others may not begin, even if they can finish on the 14th. The three permitted professions, i.e. tailors, barbers and launderers, may work until midday. Other professions, if they already started before the 14th, may finish until midday.

Beis Yosef (DH v'Chasav): The Rambam holds like the Rosh, that R. Meir permits only until midday. The Ran says that R. Meir permits even where the custom is not to work. We conclude that if the custom is not to work, one may not begin any Melachah, even if needed for the Mo'ed. Where the custom is to work, until midday one may even begin.

Magen Avraham (13): One may not shave after midday even through a Nochri. I say that it is because he tilts his head to help. The Bach says that Maharshal permits through a Nochri, but not through a Yisrael, even for free. The Levush connotes that the custom is to forbid even through Nochrim. Maharil and Mahariyo forbid cutting nails.

Mishnah Berurah (25): Even for needs of the Mo'ed, the Heter is only if he already began, or the three professions.

Rema: Some are lenient to permit three professions to begin and work until midday even where the custom is not to work. If other professions began during the day, and it is needed for the Mo'ed, they may work until midday. This is the custom.

Gra (DH v'Yesh): This opinion explains that the Mishnah discusses a place where the custom is not to work. "We may do" is even not for needs of the Mo'ed. The Gemara derived that the Beraisa permits only where the custom is to work, but not where it is not to work. According to the Rambam (the first opinion), to answer the question (about the Mishnah) there was no need to derive the law where the custom is not to work!

Kaf ha'Chayim (69): The Rema is like the Ra'avad, Rosh, Tur and Ran.

Mishnah Berurah (29): Where the custom is to work, one may begin any Melachah, and finish even not for needs of the Mo'ed.

Mishnah Berurah (30): During the day means before sunrise of the 14th.

Mishnah Berurah (32): Once midday comes, one must stop in the middle of the Melachah.