The famous Halachic rule states: "Hadlaka Osah Misva"-The Misva of Hanukah candles is fulfilled immediately upon lighting. Therefore, "Kavta- En Zakuk La"-If the candles inadvertently blew out immediately after being lit, it is not obligatory to relight them, as long as they had enough oil to potentially burn the requisite time. This applies not only to lighting on weekdays, but even Ereb Shabbat, when they blew out before dark.
If one extinguished the candles on purpose, he should relight without a Beracha. If they blew out inadvertently, and one wants to be stringent and relight them, Hacham Ovadia rules Tavo Alav Beracha, it is praiseworthy. This is especially true on Ereb Shabbat, when some Poskim rule that by law he is obligated to relight.
If on the sixth night, the first candle went out before he lit the sixth, The Biur Halacha states that although he lit a total of six candles, it is not considered "Mehadrin Min HaMehadrin"-the ideal way to fulfil the Misva, since the six candles were not all lit at the same time.
The Mishna Berura rules that "Kavta En Zakuk La" applies only if the candles were lit in a place that was suitable for lighting. Therefore, if it was windy at the time of lighting and the candles became extinguished, they must be relit. Hacham Bension discusses whether this applies to those who light outside in the special glass boxes. The boxes do have a glass cover that prevents the wind from entering. On the other hand, at the actual time of lighting the box is open and the wind can enter. He rules that even if the candle blew out while the box was still open, he does not have to relight it, since the box is designed to be outside.
Even if the candles blew out inadvertently immediately after lighting, it is not required to relight them, even on Ereb Shabbat. If one wants to relight them, it is praiseworthy. If one blew them out on purpose, he must relight, albeit without a Beracha. If the candles blew out from the wind, while being lit in the special glass box outside, it is not required to relight them.