Shemini Atzeret
The eighth day of Succot is known as Shemini Atzeret. In Israel, we combine this day with Simchat Torah, the rejoicing with the Torah.
Unlike the last day of Pesach, where we do not say the שהחיינו blessing in Kiddush or candle lighting, we do say שהחיינו on Shemini Atzeret.
The Rabbis refer to it as a חג בפני עצמו, a holiday in itself. Hashem added this extra holiday because he said, קשה עלי פרידתכם, it is difficult for me to part with my beloved people, during the long winter ahead. In essence, He is asking His people to stay another day.
This day is somewhat of a paradox. On the one hand, it is connected to Succot, as the obligatory Chagiga sacrifice can still be offered on the eighth day, Shemini Atzeret. But there is a mnemonic that demonstrates that it is its own holiday.
The mnemonic has the letters פז״ר קש״ב, as a way of remembering these differences. The letter פ stands for פיס, which is where lots were drawn among the Kohanim to see who gets to offer the one bull of the day. During Succot, seventy bulls are offered during the seven days of Succot. Thirteen bulls the first day, twelve the second, etc.
The second letter ז refers to זמן. This is a reference to the blessing of שהחיינו recited on שמיני עצרת.
The letter ר refers to the term that it is a רגל בפני עצמו, a holiday in itself.
The ק refers to the word, קרבן, in that the holiday sacrifice is different on this day.
The ש refers to the שיר של יום, song of the day sung by the Leviim on this day. And the ב refers to the special ברכה, blessing that was given to the king from the time of King Solomon.
We see that Shemini Atzeret is a very unique day in the Jewish calendar with very unique features.