SHEMINI ATZERET begins this evening at sundown. In Israel, this is also the holiday of Simchat Torah (Simhat Torah). Outside of Israel, that holiday begins tomorrow night at sundown.
Shemini Atzeret is a biblical holiday mentioned in Numbers 29:35-38. Simchat Torah is not a biblical holiday, but it marks the beginning and ending of the annual cycle of Torah readings.
Best wishes to all those celebrating.