Normally, when we recite Habdala on Mosa’eh Yom Kippur, we recite the Beracha of "Boreh Me’oreh Ha’esh" over a "Ner She’shabat" – a candle which had been burning since before Yom Kippur. We light a 26-hour candle before the onset of Yom Kippur, and then use that flame for the Beracha of "Esh" during Habdala after Yom Kippur. This practice is followed even when Yom Kippur falls on Shabbat. Despite the fact that on Mosa’eh Shabbat we generally light a new flame to commemorate the discovery of fire by Adam Harishon, nevertheless, when Mosa’eh Shabbat is also Mosa’eh Yom Kippur, we use a preexisting flame.
However, Hacham Ovadia Yosef rules that if one does not have access to a "Ner She’shabbat" when Mosa’eh Yom Kippur is also Mosa’eh Shabbat, then he may recite the Beracha of "Esh" over a new flame. And thus although one should use a "Ner She’shabat" like on an ordinary Mosa’eh Yom Kippur, if one does not have access to such a flame – which often happens – he may light a new fire and use that flame for Habdala. This ruling appears in Hazon Ovadia (p. 384), Yehave Da’at (1:63), and Yalkut Yosef (p. 423; listen to audio recording for precise citation).
Besamim are not used in Habdala on Mosa’eh Yom Kippur, even when it falls on Mosa’eh Shabbat. Although we generally use Besamim in Habdala on Mosa’eh Shabbat, this is not the case when it is also Mosa’eh Yom Kippur. Hacham Ovadia Yosef writes that it is preferable to make a Beracha over Besamim and smell them after reciting Habdala and drinking the wine, but this should not be done as part of Habdala.
One may not eat on Mosa’eh Yom Kippur before reciting or hearing Habdala. Even once it has become dark and Yom Kippur has ended, one may not eat before Habdala. Although the other prohibitions of Yom Kippur no longer apply once it has become dark – such as bathing, wearing leather shoes and applying creams – eating and drinking remain forbidden until Habdala. However, somebody who is very thirsty may drink water before Habdala (even if he has yet to recite Arbit), just as on Mosa’eh Shabbat.
A woman whose husband has not yet returned from the synagogue after Yom Kippur may recite Habdala so she can eat.
If one has access to a preexisting flame, but it will take time for somebody to bring it, he may recite Habdala without the Beracha of "Esh" so he can eat, and then recite the Beracha of "Esh" later, when the flame arrives.
Summary: When Yom Kippur falls on Shabbat, the Habdala recited after Yom Kippur is the same as Habdala on an ordinary Mosa’eh Yom Kippur – Besamim are not used, and the Beracha of "Esh" is recited specifically over a fire that had been lit before Yom Kippur. However, when Mosa’eh Yom Kippur is also Mosa’eh Shabbat, one who does not have a candle that was lit before Yom Kippur may recite the Beracha over a new flame. Once Yom Kippur ends, all the activities forbidden on Yom Kippur become permissible, except for eating and drinking, which remain forbidden until Habdala, though one who is very thirsty may drink water before Habdala. A woman may recite Habdala before her husband returns from the synagogue so she can eat.
