Jewish communities in the Diaspora observe the eighth day of Sukkot as Shemini Aseret, and the ninth day as Simhat Torah. This is in contrast to the custom observed in Israel, where Shemini Aseret and Simhat Torah are observed together on the same day, the eighth day of Sukkot.
It is customary to perform Hakafot on Simhat Torah, both in the night and during the day. Some Diaspora communities, however, have the custom to perform Hakafot on Shemini Aseret, as well. These Hakafot are generally held on the afternoon of Shemini Aseret, after Minha.
The Hid"a (Rav Haim Yosef David Azulai, 1724-1806) disapproved of this practice, and held that it is improper to conduct Hakafot on Shemini Aseret. He wrote that if people perform Hakafot on Shemini Aseret, they will treat the Simhat Torah observance lightly, and not afford it proper respect. People might figure that since they already conducted Hakafot on Shemini Aseret, there is no need to celebrate on Simhat Torah. The Hid"a was therefore opposed to conducting Hakafot on Shemini Aseret, and insisted that Hakafot be held only on Simhat Torah.
Others, however, disagreed. The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) wrote that nowadays there is, thank God, a good deal of awareness of and respect for Simhat Torah, and we therefore need not be concerned that people will disregard the Simhat Torah celebration. In his view, it is permissible and in fact admirable to conduct Hakafot on Shemini Aseret. This was also the opinion of Rav Yaakob Haim Sofer (1870-1939), in his Kaf Ha’haim (669:33). This is the view accepted by Hacham Ovadia Yosef, who ruled that it is entirely permissible to conduct extra Hakafot on Shemini Aseret, in addition to the regular Hakafot on Simhat Torah.
Summary: It is customary to conduct Hakafot on Simhat Torah, both at night and by day. Nevertheless, there are communities that perform Hakafot also on Shemini Aseret, and although some authorities disapproved of this custom, the accepted Halacha is that this is a perfectly legitimate practice.