The day after each of the three Holidays is called Isru Chag. The name Isru Chag derives from the verse “Isru Chag Baavosim Ad Karnei Hamizbeiach”. This means to say that this day is to be attached [i.e. Isru] to the Holiday itself, and by doing so the verse considers him to have built an Altar and sacrificed on it an offering. For this reason, the following customs are relevant on Isru Chag:
Increasing in eating and drinking on Isru Chag:
One is to increase a little in eating and drinking on Isru Chag, the day after each of the three festivals.
Fasting on Isru Chag:
It is forbidden to fast on Isru Chag, the day after each of the three festivals. Even a Chasan and Kallah which are getting married on Isru Chag [as they follow the customs of Sefirah only after Rosh Chodesh] are not to fast that day. Similarly, a child may not fast on his parent‘s Yartzite that falls on Isru Chag.
One is to increase a little in eating and drinking on Isru Chag. Even a Chasan and Kallah on the day of their wedding may not fast on this day. Similarly, a child may not fast on his parent‘s Yartzite.
Do the customs of Isru Chag apply also on the night after [i.e. Motzei] Isru Chag?
Is one to wear Shabbos clothing on Isru Chag?
Sparks of Kabalah:
The Arizal taught that on the day after Yom Tov, Isru Chag, a ray of the Holiday still shines.
 Admur 429:17
 Admur 429:17; Sukkah 45b
 Admur ibid; Rama 429:2
 Admur 429:17; Rama 429:2; Sukkah 45b based on second explanation of Rashi ibid [according to his 1st explanation, the Mitzvah is to increase on Yom Tov itself, and not the next day]
 The reason: Anyone who attaches [Lit. Issur which means bound] the day after the festival to the festival itself with eating and drinking, meaning through increasing slightly in eating and drinking the day after the Holiday, and thus makes that day attached [Lit. Tafal which means secondary, or attached] to the Holiday itself, the verse considers him to have built an Altar and sacrificed on it an offering. This is based on the verse that states “Isru Chag Baavosim Ad Karnei Hamizbeiach”. Meaning to say that when one makes an Issur, a secondary day, to the festival, then Baavosim, it is considered as if he brought large and fat animals to the altar. For this reason the custom is in these provinces to increase a little in eating and drinking on the day after each of the three festivals. [Admur ibid; Sukkah ibid]
Other reasons: Some write that the celebration of Isru Chag began in Eretz Yisrael in order to show some sign of festivity on the second day of the festival of the Diaspora. This then spread to the Diaspora itself, on their Isru Chag. Alternatively it corresponds to the sacrifices which were able to be eaten for two days and one night. [Sdei Chemed Kelalim Alef 154] Alternatively it is in memory of the pilgrimage which would return home on Isru Chag. [Glosses of Chasam Sofer 429]
 Admur 429:17; M”A 429:8
 Custom or prohibition? The above prohibition however is only a custom, however from the letter of the law there is no prohibition to fast, although one who refrains from doing so is praised. [Admur ibid; M”A 429:8] This however only applies to the day after Pesach and Sukkos, however on the day after Shavuos from the letter of the law it is forbidden to fast. [Admur 429:18] The reason for this is because on the night of Isru Chag of Shavuos all the sacrifices of the pilgrimage were offered in the Temple, and it was thus made a festival. [494:19; Seemingly according to this also Erev Pesach should be forbidden from the letter of the law, being that all the peach sacrifices were brought then. However in 429:10 it is not mentioned in the list of days that are prohibited from the letter of the law to fast. Vetzaruch Iyun. The practical ramification is in whether one may make up a Taanis Chalom on that day.]
A Taanis Chalom: It is certainly permitted to fast a Taanis Chalom on Isru Chag, as even o9n Shabbos it is permitted. Nevertheless Tzaruch Iyun if such a fast requires a second fast as a Kaparah.
 Admur ibid; M”A 573:1
 Admur ibid; Rama 429:2 regarding all days of Nissan; See also Rama 568:9
Fasting on the Yartzite of parents on other days in Nissan: From here it is implied that one may fast on a Yartzite on the remaining days of Nissan. This is unlike the Rama 429:2 which rules a Yartzite fast may not be done at all during Nissan. Admur in 429 omits this ruling of the Rama.
 Ashel Avraham of Butchach 429
 The reason: As this is similar to Kodshim in which the night follows the day in terms of the burning of the offerings from the sacrifice. [ibid]
 Torah Leshma 140; Darkei Chaim Veshalom 524
 The reason: This is done in order to actively show the continued Holiness of the festival that is relevant to this day, and so one does not treat it like a regular weekday. [Torah Leshma ibid]
 Torah Leshma 140