Women light candles on both nights of Rosh Hashanah. On the second night, the candles must be lit only after nightfall, once the first day of the holiday has ended, as it is forbidden to make preparations on the first day of Yom Tob for the second day. The common custom is to wait until 40 minutes after sunset on the second night of Rosh Hashanah before lighting candles.
There are those who, every Shabbat and Yom Tob, refrain from Melachot De’Orayta – activities proscribed by force of Torah law – until 72 minutes after sundown, in deference to the view of Rabbenu Tam (Rav Yaakob Tam, France, 1100-1171) that the day ends only at that point. Such people should wait until 72 minutes after sundown before lighting the candles on the second night of Yom Tob, because this, too, involves a Melacha forbidden by Torah law. (They may, however, heat food for the Yom Tob meal already 40 minutes after sundown.)
Although kindling a flame is permitted on Yom Tob, this is allowed only from a preexisting flame; it is forbidden to create a new flame, such as by striking a match, on Yom Tob. It is therefore customary to light before Rosh Hashanah a flame that will remain kindled throughout Rosh Hashanah, so that one may light fire whenever it is needed. Hacham Bension Abba Shaul (Israel, 1924-1998) ruled that it is forbidden on Yom Tob to create a flame even by placing a match on a hot surface, as kindling is allowed only from a preexisting flame.
It is also forbidden to extinguish fire on Yom Tob, and therefore, after lighting the Yom Tob candles on the second night, the woman must put the candle or match with which she lit on a metal surface and allow it to extinguish by itself.
The custom of women in our community is not to recite "She’hehiyanu" when lighting candles on Yom Tob, and to instead fulfill the requirement of "She’hehiyanu" by listening to its recitation at Kiddush.
Summary: Candles should be lit on the second night of Rosh Hashanah no earlier than 40 minutes after sundown. Those who generally refrain from Melacha after Shabbat until 72 minutes after sundown (in deference to the view of Rabbenu Tam) should ensure not to light until this point on the second night of Yom Tob. The candles must be lit from a preexisting flame, and therefore one must ensure to light a long-lasting flame before Yom Tob. One may not extinguish a flame on Yom Tob, and so after lighting the Yom Tob candles on the second night, the woman must put the candle or match with which she lit on a metal surface and allow it to extinguish by itself.