The obligation of Hadlakat Nerot, the Ereb Shabbat candle lighting, requires lighting at least one candle, but it is customary for women to light two candles. The most common reason given for this practice is that the two candles correspond to the two commands of "Zachor" and "Shamor" ("remembering" and "observing" the Shabbat).
Additionally, the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) explained that this practice alludes to the theme of Shalom Bayit (peaceful relations between husband and wife) which underlies the Misva of Hadlakat Nerot. The Sages comment that candle lighting is required with the onset of Shabbat because the presence of light helps ensure Shalom Bayit, as darkness often leads to tension and arguments. Lights must be kindled in the home before Shabbat begins so that an aura of peace and tranquility will pervade the home. This concept, the Ben Ish Hai notes, is alluded to in the two candles that the woman lights. The Hebrew word "Ner" ("candle") has the numerical value of 250, and thus two candles have a combined numerical value of 500. There are 248 "Ebarim" (limbs and organs) in a man’s body, and 252 in a woman’s body, which together combine to form a total of 500. Thus, the lighting of two candles alludes to the "togetherness" of the husband and wife, which is represented by the number 500, and which constitutes the fundamental reason behind the Misva of the Shabbat candles.
The time of lighting the Shabbat candles is an "Et Rason," a moment when a woman’s prayers are more readily accepted by God. It is therefore proper for a woman when lighting the Shabbat candles to offer a prayer for Shalom Bayit, for her husband’s success in earning a living, and for the well-being of her children, particularly that they should grow to become Torah scholars.
The custom among the Sepharadim is that only the mother lights the Shabbat candles; single daughters do not light according to our custom. If the mother is away for Shabbat, such as if she had a baby and must spend Shabbat in the hospital, then the husband should light the Shabbat candles in her stead. He recites the usual Beracha of "Le’hadlik Ner Shel Shabbat." If both parents are away for Shabbat and their children remain home, and among the children is a daughter above the age of Bat Misva, then she should light the Shabbat candles in the home with a Beracha. Hacham Ovadia Yosef rules that a daughter above the age of Bat Misva takes precedence over a son over the age of Bar Misva, even if the boy is older than the girl. If there is no girl over the age of Bat Misva, then a boy who is over the age of Bar Misva should light the candles, with a Beracha.
Of course, the parents must light Shabbat candles in the place where they spend Shabbat. But if they have children staying at home for Shabbat, then either a son or a daughter must light the Shabbat candles, as discussed.
Summary: It is customary for women to light two candles before the onset of Shabbat, and the time of Shabbat candle lighting is a time especially suited for praying for the well-being of oneself, one’s husband and one’s children. If the mother is away for Shabbat, then the husband lights the Shabbat candles. If both parents are away for Shabbat but the children remain home, then the candles should be lit by a girl above the age of Bat Misva. If there is no girl this age, then a boy above the age of Bar Misva should light the candles.
