Shemini Atzeret is the final day of festivities at the end of Sukkot. Shemini Atzeret literally means "the assembly of the eighth (day)." Rabbinic literature explains the holiday this way: G-d is like a host, who invites us as visitors for a limited time, but when the time comes for us to leave, He has enjoyed himself so much that He asks us to stay another day.
Shmini Atzeret in 2015 is on Monday, the 5th of October.