The laws for lighting candles on Sh’mini Atzeret are similar to those for Shabbat. These laws also apply to Simḥat Torah.
The exception is that the custom of the waving of one’s hands over the flames and covering the eyes while the blessing is recited is not done as on Shabbat, because a flame can be transferred on Sh’mini Atzeret as on the other festivals, and there is no need to have lit the candles before reciting the blessing.
The blessing is:
בָּרוּךְ אַתָּה ה׳ אֱלֹהֵֽינוּ מֶֽלֶךְ הָעוֹלָם אֲשֶׁר קִדְּשָֽׁנוּ בְּמִצְוֹתָיו וְצִוָּֽנוּ לְהַדְלִיק נֵר שֶׁל יוֹם טוֹב.
Barukh attah adonai, eloheinu, melekh ha-olam, asher kidd’shanu b’mitzvotav v’tzivvanu l’hadlik neir shel yom tov.
Praised are You, Adonai, our God, Sovereign of the universe, who, sanctifying us with divine commandments, has commanded us to kindle the festival lamp.
This is followed by the She-heḥeyyanu (also written as shehechianu) blessing, which is recited on Sh’mini Atzeret and Simḥat Torah:
בָּרוּךְ אַתָּה ה׳ אֱלֹהֵינוּ מֶלֶךְ הָעוֹלָם שֶׁהֶחֱיָנוּ וְקִיְּמָנוּ וְהִגִּיעָנוּ לַזְמַן הַזֶּה.
Barukh attah adonai, eloheinu, melekh ha-olam, she-heheyyanu, vekiyamanu, vehigiyanu lezman hazeh.
Praised are You, Adonai, our God, Sovereign of the universe, who has given us life, sustained us, and brought us to this occasion.
In addition a special Yizkor candle is lit just prior to the formal Sh’mini Atzeret candles by those who will be remembering loved ones no longer living.
There is no blessing recited on the kindling of this candle, although many prayerbooks include appropriate devotional material to recite just before lighting the memorial candle.
The same blessing and procedures are used for lighting candles on Simḥat Torah, with the exception of lighting a Yizkor candle.
Adapted with permission from The Observant Life.