Asru Chag

Where it appears in the Shulchan Aruch: The halachos of asru chag (the day right after a Yom Tov) are not gathered in one siman; they come up incidentally. In this volume they appear in two places: once at the beginning of the laws of Pesach, and once in the laws of Shavuos. It therefore makes sense to learn the topic before the laws of sefiras ha'omer, because although it is stated regarding Pesach, it has a bearing on Shavuos as well.

Shulchan Aruch, siman 429, se'if 2, at the end of the se'if:

"And the custom is to increase somewhat in eating and drinking on the day after the festival, which is asru chag."
(Shulchan Aruch)

(Some call the day after Pesach "Mimouna," but Mimouna is observed only once a year, whereas asru chag applies to the other festivals as well.)

The plain meaning of the verse: "Isru chag ba'avosim"

Neither the Shulchan Aruch nor the Mishnah Berurah explains the expression "asru chag," and that is not their way. The phrase is taken from the verse we say in Hallel:

"Bind the festival offering with cords, up to the corners of the altar."
(from Hallel)

Three korbanos for the festival: One who comes up to Yerushalayim for the festival is obligated to bring three types of korbanos: shalmei chagigah, shalmei simchah, and olas re'iyah. The first two come because of the festival, and part of them is eaten by the owner, since there is no simchah without meat and wine. The olas re'iyah comes because he enters the Azarah and appears before the Holy One, Blessed be He; it is entirely for Hashem and is completely burned.
The verse according to Rashi: "Chag" means the festival offering, whether the chagigah or the shalmei simchah. "Isru" means they tie it up, to the legs of the bed or with a rope, and keep it tied until the festival, when he will go up to Yerushalayim and offer it on the altar.
Why tie it up: The korban must be free of any blemish, so one does not wait until the last minute. Rather, the animal is prepared several days in advance and watched so that it does not become blemished, so that it will arrive in Yerushalayim intact and be offered as required.
On the plain level there is no connection to the day after the festival: The verse speaks about the day of the festival itself, when the korban is brought, and says nothing about the following day that we call asru chag.

The derashah of Chazal: "Make a binding for the festival"

The wording of the Gemara: The Gemara does not use the phrase "asru chag" but rather "make an issur (a binding) for the festival," meaning: extend the time of the festival and continue it one more day, a day on which there will be certain things forbidden, a sort of festival.
A parallel to Shemini Atzeres: The idea recalls Shemini Atzeres, about which the Holy One, Blessed be He, says, "Your parting is difficult for Me," and the staying together continues one more day.
The wording of the Yerushalmi: The Yerushalmi does not mention the term "asru chag" at all; it calls this day "b'rei d'mo'ada," the son of the festival. Just as with the melachos of Shabbos there are avos (primary categories) and toldos (derivatives), which are like children, so too there is the festival itself and there is something derived from the festival, which is not the festival itself.
Why specifically one day: The reason the festival is extended by exactly one day is not explained here, and the question remains open.

The halachos of asru chag in the Mishnah Berurah

"The custom is not to fast on any asru chag."
(Mishnah Berurah)

The prohibition of fasting: Strictly speaking, once the festival is over, fasting would be permitted; the halachah comes to teach that one does not fast on asru chag, since this day still has a piece of the festival in it and a certain festive significance.
Tachanun: Some added that Tachanun is not said on it, but on the asru chag of Pesach that is no novelty, since Tachanun is not said the entire month of Nisan. The practical novelty is the prohibition of fasting.

Declaring a niduy in shul in Nisan and in Tishrei

"And as for declaring a ban in the shul, for a great need it is permitted in Nisan, even though they forbade it in Tishrei."
(Mishnah Berurah)

What a shul assembly was: When there were disputes in the community, the public would be gathered in the shul, usually before the reading of the Torah, and the wrong would be publicized before everyone.
A public announcement: The injured party himself gets up and announces that he lent money to his fellow, that months have passed and he has not repaid, and that the beis din ruled in his favor.
Declaring a niduy: A more severe option, in which one of the shul board, the community council, or the beis din gets up and announces that it is forbidden to speak with the debtor, forbidden to give him an aliyah, and forbidden to sit next to him. This is not a full cherem but a certain form of distancing from the community, and it was one of the main tools a beis din had in the diaspora to force the debtor to pay.
The authority of the beis din: There were countries where the authorities gave the Jews stronger backing, and the beis din could even impose prison sentences. The discussion here concerns a situation where they have no such authority, and the only instrument left to them is niduy.
Why it is forbidden in Tishrei: Tishrei is a month in which we pray for forgiveness of sins, and a public announcement that so-and-so owes money and is not paying arouses accusation against Klal Yisrael. Therefore one does not initiate this in Tishrei; rather, one waits until the month passes.
Why it is permitted in Nisan, and only for a great need: Nisan is not a month of teshuvah, so it does not carry the concern that Tishrei does. Still, Nisan too is a special month: in Nisan they were redeemed and in Nisan they are destined to be redeemed, and we do not want to arouse accusation that would delay the redemption. Therefore it was permitted only for a great need, and there is no precise definition of what constitutes a great need.

The second place: the night and day after Shavuos

Shulchan Aruch, siman 494, se'if 3:

"It is forbidden to fast on motzaei Chag HaShavuos."
(Shulchan Aruch)

"And we do not say Tachanun from the beginning of Rosh Chodesh Sivan until the eighth of the month, that is, after asru chag."
(the Rema)

The change in wording: The Shulchan Aruch does not call this day "asru chag" at all, but rather "motzaei Chag HaShavuos," while the Rema calls it asru chag.
The practice today: We do not follow the Rema's wording regarding Tachanun; rather, we are stricter and do not say Tachanun until the twelfth of Sivan, all the days of tashlumin (the make-up days) of the festival.

The festival korbanos and when they are brought

On Yom Tov there is no problem: Shavuos is one of the three festivals, and one who comes up for it is required to bring shalmei chagigah, shalmei simchah, and an olas re'iyah, and to bring them all the way to Yerushalayim without their becoming blemished. On Yom Tov itself all three may be offered, and he may eat his portion just as he eats at home.
On Shabbos an individual's korban is not offered: Even someone who became obligated in a chatas, part of which is eaten by the kohanim, cannot bring it on Shabbos. An individual's korban is not offered on Shabbos at all.
Communal korbanos on Shabbos: The Beis HaMikdash functions on Shabbos with the communal korbanos. Besides the daily olas tamid, a lamb in the morning and a lamb in the afternoon, on Shabbos and on Yom Tov there is a mussaf offering, and on Shabbos it is "two yearling lambs without blemish."
The Mussaf prayer: Mussaf was instituted as a remembrance of the korban, and therefore we recite in it the verses of that day's korban. On Pesach we say the same passage all seven days, since the korban is identical each day, while on Sukkos the verses change from day to day.
Tashlumin on Chol HaMoed: If the first day of Yom Tov falls on Shabbos, nothing is lost, since the festival korbanos have make-up days during the remaining days of the festival, and he brings them on Chol HaMoed, which is also part of the festival period. The same is true on Pesach and on Sukkos.

Shavuos falling on Shabbos and the "yom tevoach shel Atzeres"

(According to our fixed calendar, Shavuos cannot fall on Shabbos. The early authorities gave a mnemonic: the days on which it can fall are Sunday, Monday, Wednesday, and Friday, hinted at in the phrase "avdu goyim," that the nations refused to accept the Torah. The discussion here is about the times of the Beis HaMikdash, when everything was fixed by the sighting of the moon, and then Shavuos could indeed fall on Shabbos.)

Seemingly no problem: If Shavuos falls on Shabbos, he will bring his korbanos on Sunday, the seventh of Sivan, since the festival has tashlumin on the following days.
The dispute with the Tzedukim: The Tzedukim read "the day after the Shabbos" literally, as a regular Shabbos, and began counting the omer on Sunday. Accordingly, for them the fiftieth day, which is Shavuos, always comes out on Sunday. This was one of the great controversies in the days of the second Beis HaMikdash, and Maseches Menachos devotes a page and a half to proofs about the nature of that "Shabbos," including the words of Rabban Yochanan ben Zakai and the proofs that it is not a regular Shabbos but Yom Tov.
The practical concern: When Shavuos fell on Shabbos, most of Klal Yisrael brought their korbanos on Sunday, and in the times of the Beis HaMikdash bringing a korban and eating it made the day a real Yom Tov for a person. The Gemara calls this day "yom tevoach shel Atzeres," the day of slaughtering the korbanos. The Tzedukim were delighted by this and would say: see, the Sages themselves are celebrating on Sunday, just as we hold.
What the Sages did: On the one hand the day was a sort of Yom Tov; on the other hand they did not want it to look like a full Yom Tov. So they deliberately permitted several things on it, among them eulogies: it is related that when an important person died on that day, one of the Sages made a point of eulogizing him, so that the Tzedukim should not say that even eulogies are not permitted on it.
The character of that day: In practice its status resembled Yom Tov, since korbanos were offered on it, they ate from them, and no work was done; but certain leniencies were practiced on it to show that it is not a full Yom Tov and not in accordance with the view of the Tzedukim.

The halachah in practice: no fasting on any motzaei Shavuos

"Because when Atzeres fell on Shabbos, the day of slaughtering the korbanos was after Shabbos, and it is like a sort of Yom Tov."
(Mishnah Berurah, s.k. 6)

The reason for the prohibition: We do not fast on motzaei Shavuos because in the times of the Beis HaMikdash they did not fast on it, since they offered shalmei simchah and ate their meat, for the joy of the festival.
A suggestion that is rejected: Some wanted to attribute this to the fact that there is no precise date for the giving of the Torah, whether it was on the sixth of Sivan or the seventh; but that is not the reason.
The difficulty: The day of slaughtering was only a Sunday, and when Shavuos did not fall on Shabbos it did not exist at all. Nevertheless the Shulchan Aruch says "on every motzaei Chag HaShavuos."
The resolution: It is not the date that determines, but the event. Since the event took place on motzaei Chag HaShavuos, we do not look at which calendar date it falls on, and every motzaei Shavuos is called a day that is like a sort of Yom Tov.
A parallel to Shabbos HaGadol: There too the miracle occurred on the tenth of Nisan, and we do not follow the date but the event, which happened on Shabbos; therefore the Shabbos before Pesach is called Shabbos HaGadol every year, regardless of the date.

In passing · Yom Tov Sheni of the diaspora and asru chag

Is Yom Tov Sheni considered asru chag: One cannot say so, because outside Eretz Yisrael that day is a full Yom Tov, not asru chag. Essentially they have no asru chag on that day, and one would bring his korbanos on it.
The source of the custom according to one view: Some say that asru chag arose in Eretz Yisrael so that there would be a certain correlation with the diaspora: since outside Eretz Yisrael that day is a complete festival, in Eretz Yisrael they made it into a sort of half festival.