Amidst his discussion of the laws of Hanukah candle lighting, the Shulhan Aruch (Orah Haim 673) codifies the principle of "Hadlaka Osa Misva," which means that it is through the act of lighting, as opposed to putting the lit candles in place, that one fulfills this obligation. The practical implication of this rule is that one fulfills his obligation by lighting the Hanukah candles even if they are extinguished shortly thereafter. (This is Halacha is called "Kabeta Lo Zakuk La.") So long as one lit the candles under conditions that would normally allow for them to burn for a half-hour period, he has fulfilled the Misva, even if they are somehow extinguished just moments later.
One does not fulfill his obligation, however, if he lights the candles in a manner that does not allow for them to burn for a half-hour. This would include failing to provide sufficient oil or wax, or kindling the flames in a location that is exposed to strong wind gusts. In these situations, if the candles are extinguished within a half-hour of lighting, one must relight them. He does not repeat the Berachot, however, as this requirement to relight the candles is subject to some debate.
As mentioned, if one lit the candles under conditions that would normally allow them to burn for a half-hour period, he has fulfilled his obligation and thus does not, strictly speaking, have to rekindle the flames if they are extinguished. It is proper, however, to be stringent in this regard and relight the candles, but one does not recite the Berachot over the second lighting. On Ereb Shabbat, too, if the candles were extinguished shortly after lighting, before Shabbat has begun, one should preferably relight them without reciting the Berachot. This is the ruling of the Taz (Rabbi David Halevi Segal, 1586-1667). If one had already accepted Shabbat, then he may ask another Jew who has yet to accept Shabbat to relight the candles on his behalf; this is the ruling of the Kaf Ha’haim (Rabbi Yaakob Haim Sofer, Baghdad-Israel, 1870-1939).
This Halacha applies even if a person accidentally extinguished the candles himself, such as if he tried to fix the wick to produce a steadier flame, and the flame was extinguished in the process. Even in this case, he is not, strictly speaking, required to relight the candle, and if he wishes to be stringent he does not repeat the Berachot. However, if a person intentionally extinguished the Hanukah candles, for whatever reason, then he must rekindle them. Even in such a case, however, he does not repeat the Berachot.
Summary: If one lit the Hanukah candles with enough oil or wax to sustain the flames for a half-hour, and in a place that would allow them to burn for a half-hour, he has fulfilled his obligation even if the candles were somehow extinguished shortly after lighting. Nevertheless, it is proper to rekindle the flames, without reciting the Berachot. If he lit the candles in a manner that would normally allow them to burn for a half-hour, he must relight them, though without reciting the Berachot.
