Besamim: Havdala is said without Besamim. In the event that Yom Kippur falls on Shabbos the blessing over Besamim is recited in Havdala as usual. [See footnote for the custom of Sefaradim]
Havdala candle: On Motzei Yom Kippur the blessing of mi’orei ha’esh may only be said over a pre-existing flame that had been lit before Yom Kippur. This applies even if Yom Kippur fell on Shabbos. [In order to have two wicks for Havdala one is to join another candle to this flame and say the blessing over both candles together.]
The flame of a gentile: One may not use the candle of a gentile that was lit on Yom Kippur.
Must the actual flame that rested be used or may one also use a candle which was lit from it? From the letter of the law any flame which was lit from a flame that was lit before Yom Kippur is valid to be used. Nevertheless it is best to be stringent to use the actual flame which was lit before Yom Kippur.
Using the candles of the Shul: One may light a candle from the Shul candles that have been lit from before Yom Kippur, and say Havdala on both candles together. If one is unable to say Havdala on both candles he is not to use the Shul candle and is rather to say Havdala over a candle that was lit from the Shul candle. Nevertheless if one used the Shul candle he does not need to repeat the blessing on a different candle.
What does one do if no pre-lit candle is available? If a pre-lit candle is not available either to use or to light another candle from it, then the blessing over fire is omitted during Havdala. This applies even when Motzei Yom Kippur falls on Motzei Shabbos.
Must one search for pre-lit candle if it is not readily available? If one does not have a candle readily available on Motzei Yom Kippur, some say he is obligated to search for a candle to say the blessing over it.
May one use a Yartzite or Teshuvah candle for Havdala?
No. However one may light another candle from them and the say a blessing.
Leaving a 24 hour candle on in house for Havdala:
One is to light a 24 hour candle in his home to be used for Havdala on Motzei Yom Kippur.
 Shulchan Aruch Chapter 624
 Other Opinions: According to the Michaber, and so is the custom of the Sefaradim Besamim is not included in Havdala even when Motzei Yom Kippur falls on Motzei Shabbos, as due to the fast there was no extra soul. [624/]
 The reason a pre-lit candle is used: Although a candle is not used in Havdala of other Motzei Yomim Tovim it is used on Motzei Yom Kippur being that on Yom Kippur it is forbidden to light candles, therefore when it becomes permitted it is befitting to make a blessing over it, as it is like a new item. Therefore only a candle which was lit from before Yom Kippur may be used even if Yom Kippur fell on Shabbos in order to show that this day is holy in that it had rested from light. It is not similar to Shabbos, as the blessing of a candle on Shabbos is said for the creation of fire on Motzei Shabbos. [ibid]
 The Shul candle is not to be used alone for the blessing over the Neir in Havdala even if it was lit from before Yom Kippur being that it is lit for honor of the Shul and not for the sake of light. [This applies especially today that the candle light serves no use for light being that we have electricity.]
 As nevertheless the candle is also made to give light. [Vetzaruch Iyun today that candles are no longer needed to give any light due to electricity.]
 So is implied from Admur 625/5; M”B 624/10; and so rules Aruch Hashulchan 624/6
Other opinions: Some Poskim rule that one may light a new flame on Motzei Yom Kippur and say the blessing over it in a time of need that no other preexisting flame is available. [Chayeh Adam 145/40; Kaf Hachaim 624/17; See Piskeiy Teshuvos 624/6]
 As rules Admur so rules also M”A 624/7. However there are some Poskim [M”B 624/7, Shaareiy Tziyon 9] which rule that on Motzei Shabbos one may use any flame if a pre-lit candle is not available. It is clear however that this is not the opinion of Admur. [See Shaareiy Tziyon ibid]
 Even if Yom Kippur falls during the week. [ibid]
 As the blessing over fire on Motzei Yom Kippur is similar to the blessing of Havdala which is said to bless Hashem for distinguishing this day of Yom Kippur over other Yomim Tovim, as on this day lighting fire was forbidden until Motzei Yom Kippur. [Admur ibid]
 M”B 298/30; Admur 624/8
 As it has been lit for the soul of the departed and not for light.
 M”B 624/13 in name of Derech Hachaim
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