atzeret

After Sukkot comes שמיני עצרת Shmini Atzeret (or Shemini Atseret or some similar spelling). We find the term first in Bamidbar 29:35:

בַּיּוֹם, הַשְּׁמִינִי--עֲצֶרֶת, תִּהְיֶה לָכֶם
"On the eighth day you will have an atzeret"

Similarly we find it mentioned in Vayikra 23:36:
בַּיּוֹם הַשְּׁמִינִי מִקְרָא-קֹדֶשׁ יִהְיֶה לָכֶם ... עֲצֶרֶת הִוא--כָּל-מְלֶאכֶת עֲבֹדָה, לֹא תַעֲשׂוּ
"On the eighth day you shall observe a sacred occasion ... it is an atzeret; you shall not work at your occupations"

The last day of Pesach is called atzeret in Devarim 16:8:
וּבַיּוֹם הַשְּׁבִיעִי, עֲצֶרֶת
"On the seventh day, an atzeret"

Additionally, in Rabbinic Hebrew, the holiday of Shavuot is known as Atzeret.

What is atzeret? It probably won't surprise you to read that there are a number of opinions:

a) Assembly, gathering: In Yeshayahu 1:13, we find the word atzara עצרה which has this meaning. Onkelos translates the word as כניש kenish - "gathering", which is related to the word בית הכנסת - beit haknesset, literally "house of gathering".

b) Stoppage of work: The gemara gives this explanation in Hagiga 9a and 18a: עצרת - עצור מעשיית מלאכה - "atzeret - stop doing work". This is the opinion of Ibn Ezra and Sforno.

c) Delay: This is Rashi's explanation, based on the Midrash where God asks us to stay with him just one day more. Radak in Sefer Hashorashim takes a similar approach, and says that those that went up to Jerusalem for Sukkot were delayed there for one more day. Rav David Tzvi Hoffman writes that while the earlier meanings are found in other books of the Bible, in the Torah עצר only means "to delay, to restrain". The modern Hebrew verb לעצור - "to arrest, to detain" is related to this meaning.

d) Conclusion: This is how the Septuagint translates the word into Greek: exodion, meaning "finale" (and related to the word "exodus".) Bula in Daat Mikra Vayikra says this meaning applies well to the last day of Pesach and Shavuot - which concludes the period of the Omer. Jastrow defines the term as "a festive gathering for the conclusion of a festive season, concluding feast".

Of the more modern scholars, Steinberg and Klein agree with "stoppage of work", while Kaddari offers "assembly".

The concept that atzeret has more than one explanation can help us understand a difference in the text of the Yaaleh V'Yavo prayer (as discussed here and here). When mentioning Shmini Atzeret, Nusach Ashkenaz says:
ביום השמיני חג העצרת הזה
while Nusach Sefard has:
ביום השמיני העצרת החג הזה

The Nusach Ashkenaz version refers to the chag of atzeret - a holiday on its own, which fits with the definitions of "assembly" or "stoppage of work".

The Nusach Sefard version, however, has "the atzeret of the chag", where chag would seem to refer to Sukkot. Therefore the atzeret mentioned would be either be the conclusion of Sukkot, or one additional delay after the seven days of Sukkot.