The name Shemini Atzeret is derived from the verse in Numbers 29:35

'On the eighth day (" shemini ") you shall hold an " atzeret " (assembly)...'

G-d finds it difficult to separate Himself from Israel, and therefore asked them to remain behind an extra day. Rav compared this to a king whose subjects and entire family came to join him and honour him when he held a celebration. At the end of the celebration, when all the subjects left, he asked his close family to remain an extra day.

All the nations of the world are expected to come to Jerusalem every year to celebrate Succot (Zechariah 14:16). When they all return home, Israel will remain an extra day in Jerusalem, Shemini Atzeret .

In the Diaspora, the day after Shemini Atzeret (which, is 'the second day of Shemini Atzeret ' - like the second day of Pesach, Shavuot and Succot) is Simchat Torah . In Israel, Simchat Torah is celebrated on Shemini Atzeret itself (i.e. both festivals are celebrated together, on one day).

Why is Simchat Torah celebrated in Israel on Shemini Atzeret ?
Because Shemini Atzeret is to Succot as 'Atzeret' (the talmudic name for Shavuot) is to Pesach:

• On Pesach, Israel saw G-d's miracles and fifty days later entered the Contract of the Torah.
• On Succot the Jews are purified from their sins, following the pardon granted on Simchat Torah, and with happiness and love enter the protection of the succah which symbolises G-d's protection. The happiness and love of G-d are expressed through the Torah.
  And how is joy expressed through Torah?
• The reading of the Torah (over the past year) is completed, and immediately begun again at the beginning, and this is celebrated with song and dance.