Leviticus 23:36

Seven days you shall offer an oblation to the Name of the Lord, you shall gather together to pray before the Lord for rain; no work of labour may you do.

for seven days you shall present a fire-offering Unlike the fire-offering of the Passover holiday, however, these fire-offerings are not all identical.

it is a stoppage [Hebrew: ‘aṣeret] Some say this word means “assembly”, as in “an assembly [Hebrew: ‘aṣeret] of treacherous men” [Jeremiah 9:1] — in that all Israel gathers together for the three festivals. This translation is incorrect, because concerning Passover Scripture says, “on the seventh day there will be an ‘aṣeret ” [Deuteronomy 16:8]; yet earlier Scripture has said, “in the morning you may return to your homes” [Deuteronomy 16:7]. It is more plausible to translate the word along the lines of “detained [Hebrew: ne‘ṣar] before God” [I Samuel 21:8] to denote a respite from wordly concerns. Accordingly, Scripture says, it is a stoppage: you will do no servile work . A similar phrase describes the stoppage on Passover.

IT IS ‘ATZERETH.’ is of the root atzor (to hold back, restrain). Rashi will accordingly explain that the eighth day of Tabernacles is the day G-d requested Israel “to hold back” yet another day before leaving for their homes. Ramban will explain atzereth in the sense of “binding,” or “holding in,” because this day represents the assembly of all emanations alluded to by all festivals, as mentioned in the text. “I keep you back before Me. It is similar to the case of a prince who invited his children to a banquet for a certain number of days. When the time came for them to leave, he said, ‘I beg of you stay with me for yet one more day, for your departure is so hard for me!’” This is Rashi’s language. And these are words of Agadah (homily) in Vayikra Rabbah.
By way of the Truth, [the mystic teachings of the Cabala], for in six days the Eternal made the heaven and the earth, and the seventh day is the Sabbath which has no partner, and the congregation of Israel is its partner, as it is said, and the earth, this being the eighth day. It is ‘Atzereth,’ for there everything is assembled. Now with respect to the Festival of Unleavened Bread He commanded that it be observed for seven days, with the first and seventh [days] being holy, though they are all holy and the Eternal is among them. From then on [i.e., beginning with the second day of Passover] we are to count forty-nine days[of the omer], which are seven weeks comparable to the [seven] “days” of the world, are comparable to “the days” of the world, and Shevuoth is thus like the “eighth day” in the Festival of Tabernacles; and the forty-nine days between the first day and “the eighth day” are thus like the intermediate days of the Festival of Tabernacles (Abusaula). and then to sanctify the “eighth day” [i.e., the Festival of Weeks] just as the eighth day of Tabernacles [is holy]; and [the forty-nine days] counted between them are in the “intermediate days” of the festival, in the interval separating the first day and “eighth day” of the festival, this being the day of the Giving of the Torah when He made them to see His great fire and His words they heard out of the midst of the fire. Therefore our Rabbis, of blessed memory, always called the Festival of Weeks by the name of Atzereth [a name here used in the verse for the eighth day of the Festival of Tabernacles], for it is on “the eighth” day of the festival, which Scripture here so called by that name. And this is the [intention of the] saying of the Rabbis: “The eighth day is a festival for itself” with respect to ‘p’zar k’shab,’ (lots), z’man (time), regel (festival), korban (offering), shir (song), brachah (blessing). These are all references to various laws indicating that the eighth day of Tabernacles is not considered as just another day of that festival, but is rather an independent festival by itself. Such acrostics were used in order to assist memory. but yet it complements the first [seven] days connected with the first seven days. since it is an emanation of the first days but is not like their unity. Therefore in the section of All the firstlings He mentions in speaking of the three festivals — the Festival of Unleavened Bread, the Festival of Weeks, and the Festival of Tabernacles — [that the Festival of Tabernacles] is seven days, 16:15. and He did not mention the eighth day, for there He said, all thy males appear etc., Verse 16. See my Hebrew commentary p. 197. This is thus clear.

עצרת, a stoppage, i.e. a day when one must refrain from work.

עצרת היא. The concept of עצירה is something apart from the concept of שביתות. The latter pertains to abstaining from ordinary secular activities, whereas the former entails spending a certain amount of time performing holy tasks, studying Torah, engaging in communal prayer, etc. The expression נעצר occurs in this context in Samuel I 21,8 ושם איש מעבדי שאול ביום ההוא נעצר לפני ה', “and there was a man, one of Saul’s officials detained before the Lord, etc.” In Yoel 1,14 we find the line קדשו יום, עצרו עצרה, “solemnize a fast, proclaim an assembly!” In Kings II 10,20 we find that Yehu proclaimed an assembly in honour of the Baal, [he meant to make sure that he could kill all those who would worship the Baal, which he succeeded in doing, eradicating this cult from Israel. Ed.] The invitation read: קדשו עצרה לבעל, “proclaim a holy assembly for the Baal.” It is in this sense that the Torah refers to the day following the seventh day of the Sukkot festival as Atzeret, an assembly for G’d, the pilgrims for the Sukkot festival to Jerusalem being detained there for an additional day. The joy on that day was to be a Torah inspired and oriented joy, akin to when David says in Psalms 149,2 ישמח ישראל בעושיו, “Israel rejoices in its Maker.”
We find that the description of the family dinners held in the home of Job’s children, his seven sons and three daughters, at set intervals such festive get togethers were held. On the day after the completion of these festivities their father would send word to them to sanctify themselves and rise early in the morning and offer burnt offerings to atone for the possibility that they might have been remiss and committed some misdemeanour which would have displeased their G’d. (Job 1,4-5)
On the seventh day after the Exodus from Egypt the Israelites took time out to sing a song of thanksgiving to G’d for their final delivery from the pursuing Egyptians. Subsequently, in commemoration of that occasion, the Torah decreed that this day become an עצרת, a day devoted to contemplation of the great miracle on that first occasion. (Compare Exodus 15,1 and Deuteronomy 16,8).
Again, on the fiftieth day after the Exodus when the Israelites in a great assembly at Mount Sinai spent the day contemplating their elevation to an almost supernatural people, this day was also subsequently observed as an עצרת, although in the Torah this day is never referred to as such. The reason why the Torah itself does not refer to this day as an עצרת may be the fact that shortly thereafter all the accomplishments of this day were lost and the Jewish people were even commanded to divest themselves of the “jewelry” they had worn in commemoration of that wonderful day. (Exodus 33,6)