When making Havdalah after Yom Kippur one is obligated to use a light ‘that rested’. This is unique to Yom Kippur, on a regular Saturday night there is no such obligation. What does this mean? And why?
A “ner sheshavas”, light that rested, requires that one light their Havdalah candle on Motzai Yom Kippur from a flame that has been burning from before Yom Kippur.
The Kol Bo explains that the candle on Motzai Shabbos and the candle on Motzai Yom Kippur are fundamentally different. The Gemara relates that on the very first Saturday Night of creation Hashem showed Adam how to make a fire from two stones. So on Saturday night in commemoration we thank Hashem for the creation of fire by lighting the Havdalah candle. It therefore may, and indeed should be, a new flame.
On Yom Kippur we are celebrating the fact that we hadn’t used fire all day. In this way Yom Kippur is different from all other Jewish holidays, on which the use of fire is permitted. So it is therefore appropriate to use a flame that had ‘rested’ all day. It had been burning before Yom Kippur, had not been used over the holiday, and had now become permitted once again.
Therefore, if Yom Kippur is on Shabbos the Poskim write that we may use a new flame, however the minhag is to use an existing flame then as well.
A candle that was lit in honor of Yom Kippur, and not for light, is a subject of dispute in Halacha. The recommended practice is to either use a candle that was lit before Yom Kippur specifically for after Yom Kippur, and not in honor of Yom Kippur. If one only has a candle that was lit in honor of Yom Kippur he should light a candle from that (which is not as bad as a brand new flame) and make the bracha on both together.
In the event that one cannot find a flame that had been burning from before Yom Kippur one should not make the blessing on the candle during Havdalah.