History of Shemini Atzeret and Simchat Torah
The status of Shemini Atzeret can be confusing at first glance. Its name means the “eighth day of assembly”--which would imply that it somehow belongs to the seven-day holiday that immediately precedes it, Sukkot. It is true that Shemini Atzeret is related to Sukkot, but its independence as a holiday is well established in the Talmud.
In Numbers 29:35 we learn that “On the eighth day you should hold a solemn gathering; you shall not work at your occupation.” This verse does not connect the 8th day specifically to any of the other traditions associated with Sukkot, begging the question: Is this its own distinct holiday or part of Sukkot? This confusion led to much debate over whether one should, for example, say kiddush (the prayer of sanctification, recited over wine on holy days) in the Sukkah on this day a custom followed by some, or whether Shemini Atzeret should warrant its own liturgical additions. [The fact that some have the custom to sit in the sukkah on Shemini Atzeret is based on the uncertainty surrounding yom tov sheni shel galuyot, that is, one would sit in the sukkah just in case Shemini Atzeret really is the seventh day of Sukkot.]
In his book The Jewish Holidays, Michael Strassfeld points out that Shemini Atzeret in many respects parallels Shavuot, which can be viewed as the long-distance conclusion to the seven-day holiday of Passover, coming as it does seven weeks after Pesach. At that time of year, the weather would be clear enough to have people come back to Jerusalem for an additional pilgrimage some weeks later. Sukkot, however, marks the beginning of the rainy season, and since it would be difficult to ask people to make an additional trip to Jerusalem, Shemini Atzeret would best be placed immediately following Sukkot. [In the Talmud, Shavuot is called "Atzeret" making the parallel with Shemini Atzeret even stronger.]
Shemini Atzeret is a two-day festival in traditional Diaspora communities and a one-day holiday in Israel and in many liberal Diaspora communities, as with many other Jewish holidays. The only ritual that is unique to Shemini Atzeret is the prayer for rain (tefilat geshem), and this prayer is parallel to the prayer for dew which is recited on Passover. These two holidays serve as the bookends of the agricultural season, at the beginning and end of the rainy season. Whereas the Torah does describe the offering that was brought to the Temple on Shemini Atzeret, once the Temple was destroyed, there was nothing that remained from the holiday’s ritual except the liturgy requesting rain for a bountiful year.