Simchat Torah is the day when Jews across the world complete the reading of the Torah and there is a celebration upon this completion. Outside of Israel, the second day of Shemini Aseret is called Simchat Torah because that's when this completion takes place.  In Israel, the one day of Shemini Aseret is also Simchat Torah.
When Hakafot take place
- In Israel there is a minhag to do Hakafot the night of Motzei Simchat Torah.
Other details of hakafot
- Even those who are strict about not dancing or clapping on Shabbat and Yom Tov all year round are lenient about dancing and clapping on Simchat Torah in order to give respect to the completion of Torah. However, playing musical instruments is absolutely forbidden on Simchat Torah, both in Israel and outside Israel.
- It is praiseworthy to engage in the celebration of Simchat Torah with dancing and clapping as much as one could. 
- The gabbai of the shul should be careful not to give a Sefer Torah to a young child to dance with because he may drop it. 
- Some say that having a person ride on someone else's shoulders is a bad minhag. 
- It is permitted for the elderly and and sickly to sit between Hakafot but they should stand during the Hakafot and when the Sifrei Torha are being returned to the Heichal. Healthy people should stand during the entire duration of Hakafot until the Sifrei Torah are returned to the Heichal. 
- On Simchat Torah, it is permitted to move a torah for the purpose of dancing or reading it somewhere else.
- One should not make Kiddush on the night of Simchat Torah before certain nightfall (i.e. Tzet HaCochavim). 
- The minhag is that everyone should receive an aliyah on simchat torah in order to be involved in the happiness of finishing the Torah. Also, it increases our honor for the Torah. 
- Some say that congregations should not say the 13 attributes of mercy (Yag Middot) when opening the aron on Shemini Aseret. 
- Kitzur S"A 138:7, Chazon Ovadya p. 467
- Chazon Ovadyah p. 469
- Chazon Ovadya p. 460
- Nitei Gavriel (Sukkot 93:9), Chazon Ovadya p. 456-7
- Chazon Ovadya p. 458
- Chazon Ovadya p. 462
- Chazon Ovadya p. 463
- Chazon Ovadya p. 463
- Chazon Ovadya p. 465
- Nitei Gavriel (Sukkot 94:1), Chazon Ovadya p. 460
- Kaf HaChaim 668:10 quoting Ben Ish Chai (Vezot HaBrachot #13), Nitei Gavriel 92:2
- Rama 669:1, Mishna Brurah 669:12. Aseh Lecha Rav 3:21 adds that this minhag is partially based on the minhag to give each person an aliyah once a month and at least in Tishrei we should ensure that this take place.
- Halichot Shlomo p. 436, Chazon Ovadya p. 469