praying for rain - tefillah Geshem

Shemini Atzeret, meaning "the eighth day of assembly," is a Biblical Jewish holiday that follows the Jewish festival of Sukkot. It is written: "On the eighth day you should hold a solemn gathering; you shall not work at your occupation".
Shemini Atzeret marks the beginning of the rainy season following the harvest in Israel.
The prayer for rain, Tefilat Geshem, is the only ritual that is unique to Shemini Atzeret. In ancient times, an offering was brought to the Temple in Jerusalem on Shemini Atzeret. But once the Temple was destroyed, the only Shemini Atzeret ritual that remained was the liturgy requesting rain for a plentiful year.
After the prayer for rain is recited on Shemini Atzeret, the phrase Masheev HaRuach U-Moreed HaGeshem (He causes the wind to blow and the rain to fall) is inserted into the Amidah prayer until Passover.
Ashkenazi Jews recite the Memorial Prayer, Yizkor, on Shemini Atzeret.

Even though Shemini Atzeret immediately follows the festival of Sukkot, it is a totally separate holiday. A new Shehechiyanu blessing is recited. And if one sits in the Sukkah on Shemini Atzeret, the prayer recited for sitting in the Sukkah is not recited.