Halacha requires that we recite Havdala on Mosa'e Yom Kippur and include in this recitation the Beracha of "Bore Me'ore Ha'esh" over a candle, just as we do on Mosa'e Shabbat. There is, however, one critical difference between the candle used for Havdala after Shabbat and the candle used after Yom Kippur. When reciting Havdala after Shabbat, we light a new candle; on Mosa'e Yom Kippur, however, one must, as the Shulhan Aruch rules (Orah Haim 624:4), recite the Beracha over a flame that was kindled before the onset of Yom Kippur and burned throughout Yom Kippur. If one does not have such a candle on Mosa'e Yom Kippur, then he does not, according to the Shulhan Aruch, recite the Beracha of "Bore Me'ore Ha'esh."
This distinction, as the Meiri (Rabbi Menahem Meiri, France, 1249-1315) explained, stems from the basic difference between the purposes served by the Beracha over the candle in the two cases. On Mosa'e Shabbat, we recite this Beracha to commemorate the creation of fire, which occurred on the first Mosa'e Shabbat after creation, when Adam rubbed two sticks together and created fire. We mark this event by creating a new fire and reciting a Beracha. On Mosa'e Yom Kippur, by contrast, we recite this Beracha for an entirely different reason, namely, to emphasize the distinction between Yom Kippur and other festivals. Whereas on all other festivals Halacha allows lighting a candle (from an existing flame), on Yom Kippur (like on Shabbat) this is forbidden. We demonstrate this difference by taking a candle that had burned throughout Yom Kippur and reciting a Beracha, as if to say that only now that Yom Kippur has concluded is it permissible to make use of the flame.
Is it permissible to use a new candle for this Beracha when Yom Kippur falls on Shabbat? Seemingly, even though the new candle obviously cannot serve as the candle for Mosa'e Yom Kippur, it still qualifies for use as the candle for Havdala of Mosa'e Shabbat. Indeed, Hacham Ovadia Yosef, in his work Hazon Ovadia – Yom Kippur (p. 378), cites a number of Rishonim (including the Ravya, Ritba, Ra'a and Meiri) who maintain that when Yom Kippur falls on Shabbat, one may use a new candle for Havdala. This view is cited as well by the Hafetz Haim (Rabbi Yisrael Meir Kagan, Lithuania, 1839-1933), in his Sha'ar Ha'siyun (624:9). Hacham Ovadia thus rules that although it is preferable to use for Havdala even in such a case a candle that had been kindled before Yom Kippur, if no such candle is available one may recite the Beracha over a newly-lit candle.
Summary: On Mosa'e Yom Kippur, one recites Havdala and includes the Beracha of "Bore Me'ore Ha'esh" over a candle. This Beracha must be recited specifically over a candle that had been lit before Yom Kippur and burned throughout Yom Kippur. If no such candle is available, then the Beracha is not recited. If, however, Yom Kippur falls on Shabbat, then one may light a new candle and use it for Havdala if he does not have a candle that was kindled before Yom Kippur.