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?
Yehoshua, Yerushalayim, Eretz Yisrael
Shemeni Atzeres celebrates the eternal bond of Hash-m and Bnei Yisrael. This is as Rashi explains in Vayikra (23:36) that it is as if Hash-m is a king who invited us for a meal for many days, and He afterwards says, "It is hard for Me to depart from you. Stay with Me for one more day." This is as the Sefer Ha'Chinuch explains (Mitzvah 324) that the reason that we do not take the Arba Minim on Shemini Atzeres is because it is a day that is totally dedicated to us being with Hash-m, and we do not need any "remembrance" on this day.
All the best,
Yaakov Montrose