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 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)
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.
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:
I see those mitzvot directly answer what we celebrate:
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: