According to the Gemorah in Sukkah Daf 48a Shmini Etzeres is a Chag in and of itself. However what exactly is the "chag" of Shmini Etzeres. Sukkos we are commemorating the Anei HaKavod, Pesach is Yetzias Metzriam, Shavuos is Matan Torah. What exactly are we "celebrating" on Shmini Etzeres?
Interestingly, the Torah itself doesn't mention anything special we are celebrating, nor does it give any specific Mitzvah to relate to (besides the special קרבן מוסף).
The Yalkut Pinchas 782(ילקוט שמעוני פרשת פנחס רמז תשפב) brings a משל (from TB Sukah 55b) describing Sukot as a party a king has with many friends. When the party is over he asks his son to stay a bit longer to have time only with him. According to this we are 'celebrating' being close to Hashem, to show his love to us.
Not to be cholek on @IsaacMoses (or Rashar for that matter), I looked up the word עֱצֶרֶת itself. It seems always to mean, according to most commentaries I've seen, something along the lines of rally/get-together/group. This goes well with what I've written previously - us having a party with Hashem.
One may note, that שביעי של פסח and שבועות use this term as well. For שביעי של פסח Yonatan Ben Uziel translated (English mine)
get together in praise to Hashem. As these times are all an end of a period, this getting together seems like a summary or חשבון נפש, resembling what @Isaac brought.
Below is what R' Hirsch says on the subject (Commentary on Lev. 23:36, on p. 684 - 686 of the original Levy translation, with italicized transliterations, elisions, and added paragraph divisions by me).
Very brief summary: The special nature of the holiday is indicated by its name "Atzeret," meaning "a preserving summing up" (R' Hirsch's translation), and the fact that it's on a separate day from the rest of Sukkot. These elements tell us that we're meant to focus on gathering all the lessons of Sukkot and the entire year's worth of holidays to help us lead the rest of our lives in the presence and service of God.
The denomination of this special mo'ed-day runs: - atzeret hee. ... If we examine the places where the root 'aTzR is used in Tanach, we find, in the far predominating number of cases, that it has the meaning of retaining, keeping back persons, things, or forces to guard against their becoming withdrawn or lost. ... We accordingly thing we are not wrong if we take atzeret to designate a day which is not fixed to bring new lessons and new truths for us to accept and assimilate, but which has the mission to keep us still before the presence of God ... to strengthen and solidify the impressions and knowledge we have already gained, so that they remain with us permanently, and do not become lost in the hurly-burly of life.
This would be entirely in accordance with the designation of the last day of a festival. Take Pesach: for six days we have lived permeating oursevles with all the facts and teachings which the yetziat-Mitzrayim-thoughts pledge and grant us, the Children of Israel. The seventh day is atzeret (Deut. 16:8), it is a gathering up of all the spiritual "treasures" that we have collected during the days of the Festival before the Presence of God, so that we can, at the end of this seventh day, step over into the course of our ordinary life which then begins, truly enriched by them.
The purpose of atzeret is accomplished by our realising once again all that we have gained by the festival, and by the firm resolution not to allow ourselves to be robbed in the turmoil and struggle and work of our lives of what we have won. Therein lies the dual conception of 'aTzR: persistance, remaining with God, and holding fast to all the spiritual acquisitions obtained, both really at the bottom coming to the same thing.
Now, it atzeret were on the seventh day of Sukot, it would only have the meaning of summing up and holding fast to the teachings which have been given to us by the Festival of Sukot. But as atzeret here comes bashemini as an independent festival on its own, and makes the mission of persevering and continuing and holding on before God, into a separate mo'ed-idea, and with it brings to a close, not only the cycle of Festivals of the seventh month, but the whole cycle of the festivals of the whole year, Shemini Atzeret would that come to tell us, once again to summarise and gather to ourselves all the thoughts and messages and resolutions which the mo'adim of the whole year have brought to us, and to resolve to persevere and hold fast to them before God. To impress them so deeply in our hearts that they become an unassailable part of ourselves which can not become lost in the course of the ordinary run of our yearly life on which we are now entering. That, in all the vicissitudes which may be awaiting us, richly endowed by the mo'adim, we may "persevere and remain" with God. ...
See the original for the parts I elided, which include discussion of some sources in Talmud and Midrash that lend some support and some challenge to this interpretation.
We can discover what we are celebrating by examining the two main actions of that day:
- changing the second Amidah blessing to "Who causes the wind to blow and the rain to fall"
- changing the number of sacrifices for the gentile nations to only one sacrifice (when sacrifices were offered in the Temple).
I see those mitzvot directly answer what we celebrate:
- Celebrating God's goodness for providing the rainy season in Israel (representing the Jewish people's nourishment, physical and spiritual)
- Celebrating the intimate relationship between God and the Jewish nation
The customs of the Sephardic Tikkun HaGeshem and Ashkenazic Tefillat HaGeshem underscore the first point. I see the establishment of Simchat Torah as related to the second point, in which the Jewish people celebrate God's gift of goodness that emphasizes this intimate relationship - that is, the Torah.
To hear from someone significant, I'd refer to Rav Adin Steinsaltz, who reflects about Shemini Atzeret this way in his book A Guide to Jewish Prayer:
Our sages see this festival as a private celebration between God and Israel. On the festival of Sukkot, we pray on behalf of the entire world, whereas Shemini Atzeret is like a "small banquet made by the King for His beloved" (Rashi on Numbers 28:35). This festival also marks the beginning of the winter season, for on it we begin to pray for the rains of the upcoming year.
Perhaps because of this special significance, it was decided to connect this festival with the ceremony for completing the yearly cycle of Torah readings and with the festivities for Simhat Torah.
The Chasam Sofer (Derashos HaShalem Shemini Atzeres VeSimchas Torah § 10, first printed in Chiddushei Chasam Sofer Sugyos (5653) § 2) says we're celebrating the completion of all the mitzvos of the month: Rosh Hashanah with Shofar, Yom Kippur with the Avodah and Teshuvah, Sukkos with the Sukkah and Arbah Minim, and of course all the korbonos.
He says a person should always be filled with joy and want to celebrate when they complete a mitzvah. He says another example is Maaser Sheni, which completes a series of agricultural mitzvos (Peah, Shichchah, Leket, Bikkurim, Terumah, Maaser Rishon). This is why the Torah tells us to rejoice when we eat it (Deuteronomy 12:7), and potentially why we read עשר תעשר on Shemini Atzeres (see Megillah 31a with Rashi). This is also why people make a siyum when they complete a Torah learning project, and why Shemini Atzeres became established as the time to complete the Torah.