One who forgot to light candles before Shekia/sunset:
The General law:
Initially, one is to light candles at the set time written in the calendar for one’s location, which is commonly at least 18 minutes before sunset. If one did not do so, then they may light the candles up until sunset. Once sunset has arrived, it is Biblically forbidden to light candles, and one who forgot to do so has lost the opportunity. One is to be very careful in this matter and is to warn his wife and family not to light the candles after sunset. Nevertheless, if a gentile is available, and it is prior to nightfall, between sunset and nightfall, the gentile may light the candle on her behalf, as explained next. Hashem will forgive the woman for missing the opportunity of lighting the candles before sunset, and she will receive a good blessing and be rewarded for abstaining from lighting the candle after sunset.
Having a gentile light on one’s behalf:
Coming up in part 2!
 See M”B 261:23; Admur in Seder Hachanasas Shabbos
The reason: Simply, the reason for this is because one must be very careful to light the candles before sunset begins, and hence it is to be lit some time before sunset. [See Admur ibid] In addition, this is in order to fulfill the requirements of all the opinions regarding Bein Hashmashos, as according to the Yereim, Bein Hashmashos begins already 15 minutes prior to sunset, hence one must accept Shabbos in their opinion, prior to 15 minutes before sunset. [M”B ibid] Alternatively, this is done because it is an obligation upon all to accept Shabbos early, prior to sunset. [See Admur 261:4]
 Siddur Admur in Seder Hachansas Shabbos; Geonim
 Letter of Admur printed as supplement to 261; See Seder Hachanasas Shabbos ibid
 Letter of Admur printed as supplement to 261