"On the eight day shall be a solemn assembly for you." [29:35]
The word atzeret is a term describing the Knesset Yisrael, the spiritual concept known as 'Israel." Were it not for the concept represented by the Jewish people, the whole universe would lack meaning and purpose. It is also an expression denoting malchut, authority, dominion. For the above-mentioned reasons Shemini Atzeret is a festival all by itself. Among the four species, it corresponds to the etrog which represents the Jewish people and which is held separately, in a different hand than the other three. The Shemini Atzeret festival is related to the festival of the giving of the Torah, meaning the festival of Shavuot, which the Talmud always refers to as atzeret. We find the name atzeret also applied by the Torah to the Seventh day of Passover (Deut. 16:6), although that day is an integral part of the festival. Thus, the term appears in connection with all three pilgrimage holidays.
What does all this mean? Just as the word Shabbat is applied by the Torah to the various festivals on various occasions to show that the Knesset Yisrael is the "bride" of the Shabbat, so the word atzeret when used in connection with the festival conveys the idea that the Jewish people, spiritually speaking, are the purpose of all these festivals. In kabbalistic terms, they are the yesod [borrowed from the emanation by that name], the "foundation," without which the entire legislation of the festivals would lack meaning.
http://www.ascentofsafed.com/c....gi-bin/ascent.cgi?Na