Shmini Atzeret is the holiday which is celebrated on the eighth day counting from the beginning of “Sukkot.” In Israel, it is celebrated for one day, which is a combination of “Shmini Atzeret” and “Simchat Torah.” In the Diaspora, “Shmini Atzeret” and “Simchat Torah” are separated, with “Shmini Atzeret” being celebrated on the eighth day, and “Simchat Torah” being on the ninth day.
Even though its celebration is keyed to the beginning of Sukkot, Shmini Atzeret is a separate holiday. Because it is considered as separate from Sukkot, the blessing of “Shehechiyanu is recited – the wife, when she lights the candles ushering in the Holiday and the husband, when he recites the Kiddush at the evening Seudah (festive meal).
“Shmini” means the eighth; in general, the number eight symbolizes perfection, as it does in the case of “brit milah,” the covenant of circumcision. The brit is performed on the eighth day to “complete,” as it were, or to instill the potential for perfection, in the human being.
“Atzeret” means “holding back,” or that which is held back for the purpose of bringing it to its state of completion. Thus, the name “Shmini Atzeret” means the eighth day which is the additional day that brings the seven-day holiday of Sukkot to its state of perfection.