The Talmud defines "night" as the period of time that begins with "Tzeit Hakochavim" (when stars are visible). The Shulchan Aruch qualifies by stating that you must be able to see 3 small stars. This is further qualified by halachic authorities who rule that the sighting of stars constitutes night only if there is no red glow that follows sunset.
So what happens in a place where the sky never darkens? The problem here is not that the day never ends, but that we cannot tell when the day has ended because there is no Tzeit Hakochavim.
Therefore:
1. You begin Shabbat at sunset (even though there is no darkening of the sky for the few minutes that the sun sinks below the horizon). Be sure to light your Shabbat candles before that!
2. Kiddush for Friday "night" needs to be recited before "halachic midnight." Halachic midnight is twelve hours after the time that the sun was directly overhead.
3. After midnight it is certainly Shabbat and the "light" is considered "morning." If you did not say kiddush by now (because the sun did not "set" below the horizon until after midnight), then you would not say it until after reciting Shacharit (the morning prayers).
4. You continue to refrain from weekday activities until Saturday night after midnight - by then Shabbat is certainly over.
5. Here again you first pray weekday Shacharit, and then recite Havdallah (the prayer said at the end of Shabbat). This is because you are not allowed to make Havdallah in the morning before Shacharit.
There was a fascinating case years ago of an Orthodox geologist working for the American government. They wanted him to do field work in Alaska for six months during the winter. He sent his question to Rabbi Moshe Feinstein, the greatest Halachic authority of the generation. Rabbi Feinstein ruled that Shabbat is twenty five hours of Sanctity and Holiness, as it is not possible to measure the time according to night and day, the geologist can determine each Friday when Shabbat was going to begin. Obviously once he had brought Shabbat in he would have to keep it for twenty five hours but the geologist could begin Shabbat each Friday at his convenience
