Shemini Atzeret ("the Eighth Day of Assembly") is a special Sabbath day that falls immediately after the seven-day Feast of Tabernacles (Sukk'ot), on Tishri 22. It is a distinct holy day, separate from the seven days of Sukk'ot itself.
It was on Shemini Atzeret — the eighth day — that Christ Jesus was circumcised according to the Scriptural command (Luke 2:21). In Hebrew tradition, a male's full birth is accomplished at circumcision, when his name is officially given. It is entirely fitting that the Son of God received His name on this special eighth-day Sabbath.
Shemini Atzeret is followed by Simchat Torah, the celebration of completing the annual Torah reading cycle.