Shemini Atzeret occurs on the day after the seven day festival of Sukkot. It falls on Tishri 22 in the Hebrew Calendar beginning at sunset the night before and is generally translated as "the eighth day of assembly".
The Talmud declares the eighth day as a separate holiday and one dedicated to the love of God. References to the celebration can be found in Bamidbar [Numbers] 29:35 ("On the eighth day you shall hold a solemn gathering; you shall not work at your occupations").
In ancient Israel, Shemini Atzeret coincided with the beginning of the rainy season. Accordingly, prayers for rain (Tefillat geshem) and good crops were recited. Many Orthodox Jews still recite the ancient prayers at this time.
Shemini Atzeret is celebrated concurrently with Simhat Torah by most Reform Jews. Orthodox and Conservative Jews celebrate the holidays separately.