An ordinance from the time of Moshe Rabbeinu mandates reading the Torah every Monday, Thursday, and Shabbat, so that three days never go by without a public Torah reading (BK 82a). Over the course of time, the custom developed to complete the Torah each year through the weekly readings (Peninei Halakha: Shabbat 5:6).
Moshe Rabbeinu further ordained that the Torah reading on the festivals would be topical, as the verse states: “So Moses declared to the Israelites the set times of the Lord” (Vayikra 23:44). The Gemara elaborates that just as Moshe spoke about the festivals, “there is a mitzva to read about each one of them at the appropriate time” (Megilla 32a; MT, Laws of Prayer 13:8). When Yom Tov or Ḥol Ha-mo’ed coincides with Shabbat, we read from the Torah about the festival instead of the regular Shabbat Torah reading. The haftara also reflects a theme of the day. On each festival, we remove two Torah scrolls from the Ark, reading the main reading from the first and maftir, in which we read about the Musaf offerings, from the second.
The number of people called up to the Torah for an aliya changes in accordance with the holiness of the day. The holier the day, the more people are called up (Megilla 21a). On weekdays, three people are called up. On Rosh Ḥodesh and Ḥol Ha-mo’ed, on which there were Musaf offerings in Temple times (and Musaf prayers today), four people are called up. On Yom Tov, when melakha is forbidden but food preparation is permitted, five people are called up. On Yom Kippur, when all forms of melakha are prohibited, six people are called up. On Shabbat, when the punishment for desecration is greater, seven people are called up. Specifically, the punishment for desecrating Yom Kippur is karet, while the punishment for desecrating Shabbat is stoning (Megilla 23a).
Thus, on Yom Tov five people are called up to the Torah, plus an additional aliya for the maftir. We may not call up fewer people than the Sages mandated, in order not to detract from the honor of Yom Tov (ibid. 21a). According to Rambam and Rashi, it is permissible to add more aliyot, but others maintain that it is forbidden, because doing so would seem to equate the sanctity of Yom Tov with the sanctity of Yom Kippur or Shabbat (Ran). Common practice follows the second opinion, for two reasons: to avoid burdening the congregation, and to avoid having extra berakhot. Originally, only the first and last people called up would recite berakhot over the Torah scroll, so calling up additional people did not involve additional berakhot. Later, the Sages ordained that each person called up recites a berakha before and after the reading. Thus if we call up additional people, additional non-mandated berakhot would be recited. Nevertheless, if there are extenuating circumstances, such as preventing a serious insult to someone, it is permitted to call up additional people (SA and Rema 282:1-2; MB ad loc. 4-5). An exception to this rule is Simḥat Torah, when the custom is to give an aliya to all men.
Yizkor: Ashkenazim have the custom to recite Yizkor, the prayer in which we remember the souls of our dearly departed relatives and pledge tzedaka on their behalf, after the Torah reading on Yom Kippur, Shemini Atzeret (Simḥat Torah in the Diaspora), the seventh day of Pesaḥ (eighth in the Diaspora), and Shavu’ot (second day in the Diaspora). During Yizkor, the two Torah scrolls are held by standing congregants. Common practice is that people whose parents are still alive leave the sanctuary before Yizkor, as it is uncomfortable to have some congregants praying and others remaining silent. Additionally, some are concerned about the evil eye. Yizkor is usually not recited for one who died within the previous year, because that might cause too much pain to the survivor and detract from his holiday simḥa.