The Laws of Candle-Lighting on Rosh Hashanah
On the first day of Rosh Hashanah, we customarily light Yom Tov candles before the onset of Yom Tov similar to the way we light them on Erev Shabbat. If the candles were not lit before the onset of Yom Tov, a woman may even light the candles on Yom Tov itself, provided that this is done in a permissible fashion, i.e. by transferring the fire from an existing flame and lighting the candles; it is prohibited to ignite a new flame on Yom Tov, as we have previously explained in the laws of Shavuot.
The Proper Time to Light Candles for the Second Day of Yom Tov
On the second day of Rosh Hashanah as well as any other second day of Yom Tov in the Diaspora (meaning the second day of Yom Tov celebrated outside of Israel for every holiday, just as two days of Rosh Hashanah are celebrated even in Israel), Yom Tov candles are lit as well.
Some have the custom to light the candles for the second day of Yom Tov of Rosh Hashanah after nightfall (i.e. approximately twenty minutes after sunset) before reciting Kiddush on a cup of wine.
The reason for the above custom is because it is forbidden to prepare anything on the first day of Yom Tov for the second just as it is forbidden to prepare anything on Shabbat or Yom Tov for weekday use. Based on this, it would seem that it is forbidden to light candles “in honor of the second day of Yom Tov” when it is still during the first day of Yom Tov.
Indeed, it is correct that one may not prepare anything on the first day of Yom Tov for the second and it is therefore prohibited to prepare the wicks, cook, or wash dishes on the first day of Yom Tov for the second for this is considered “preparing.”
Nevertheless, regarding candle-lighting, Maran Rabbeinu Ovadia Yosef zt”l writes (Chazon Ovadia-Yamim Nora’im, page 182) that based on the words of the Tosafot and other great Poskim, one need not be concerned about the prohibition of preparing and one may light candles for the second day of Yom Tov even before sunset of the first day of Yom Tov, for this serves a purpose for the first day of Yom Tov as well since the candles add light to the house. He quotes Rabbeinu Yosef Haim zt”l (in his Responsa Rav Pe’alim, Orach Chaim, Volume 4, Chapter 23) as ruling likewise.
However, this year (Rosh Hashanah 5781), when Rosh Hashanah falls out on Shabbat and Sunday, it is certainly forbidden to light candles for the second night of Yom Tov before the conclusion of Shabbat printed in calendars. Those who customarily wait for Rabbeinu Tam’s nightfall to end Shabbat throughout the year should wait to light candles until this time on the second night of Rosh Hashanah as well.
Summary: Candles are lit in honor of Rosh Hashanah just as they are lit in honor of every Yom Tov: Sukkot, Pesach, and Shavuot. The candles should preferably be lit before sunset. Candles should night be lit on the second night of Rosh Hashanah until the conclusion of Shabbat.
The great Rishon Le’Zion Shlit”a writes (in his Yalkut Yosef- Yom Tov, page 506) that a woman who usually lights with wax candles should prepare them before the onset of Shabbat/Yom Tov since melting the wax on the bottom of the candle and sticking it to the candle-stick constitutes the prohibited work of smoothing.