Halacha requires lighting the Hanukah candles in a manner that allows them to burn for at least a half-hour. If a person wishes, he may extinguish the Hanukah candles after they have burned for a half-hour. For example, if the family is leaving the home and they do not want to leave the candles burning unintended, it is entirely permissible to extinguish the candles before leaving.
Similarly, it is permissible to make personal use of the Hanukah candles after they had burned for a half-hour. Within the half-hour period, one may not use the candles for any purpose, even for a Misva – such as studying Torah by the light of the candles – or a very quick task – such as counting one’s change by the light. After the half-hour period, however, this is permissible. Some authorities allow using the Hanukah lights for any purpose at all after a half-hour, even using the flame to light the pilot in his stove. However, the Bah ("Bayit Hadash" commentary by Rabbi Yoel Sirkis, 1561-1640) maintains that after a half-hour one may use the Hanukah candles only for the purpose of performing a Misva. This is the ruling of the Mishna Berura (Rabbi Yisrael Meir Kagan, 1839-1933), and one should indeed be stringent in this regard.
Strictly speaking, one may light one Hanukah candle from another. On the second night, for example, one may, after lighting the first candle, light the second candle directly from the flame of the first. However, the Rama (Rabbi Moshe Isserles, Poland, 1525-1572) writes that it is customary not to light one Hanukah candle from another. Indeed, we customarily use a separate candle (the "Shamosh") to light all the Hanukah candles.
Summary: After the Hanukah candles have burned for a half-hour, one may extinguish them, or use them for the purpose of a Misva. Different customs exist as to whether one may use the previous night’s wicks for Hanukah candle lighting. It is customary to use a "Shamosh" to light all the Hanukah candles, and not to light one Hanukah candle from another.
