Generally speaking, one may not ask a gentile on Shabbat to perform an activity that is forbidden for Jews on Shabbat. Thus, for example, it is forbidden to ask a gentile to turn on a light, or to drive to pick up something for him.
There are, however, a number of important exceptions to this rule. Firstly, one may ask a gentile to turn on a light if there is a child present who is afraid of the dark and inconsolably frightened. Halacha allows asking a gentile to perform a Melacha (activity forbidden on Shabbat) if this is necessary for the care of an ill patient, and all children are considered "Holim" ("ill") in this regard. For the needs of a frightened child, then, it is permissible to ask a gentile to perform a Melacha even an act that is forbidden for a Jew on Shabbat by force of Torah law. Hence, it is permissible to ask a non-Jewish housekeeper, for example, to turn on a light if a child is in distress due to the dark.
Asking a gentile to perform Melacha on Shabbat is also allowed in the case of cold conditions. Halacha considers all people "ill" with regard to cold temperatures, due to the potential to fall ill from exposure to the cold. Therefore, it is permissible to ask a gentile to turn on the heat if the home is cold. Of course, one should make a point of turning the heat on before Shabbat. But if somebody forgot to turn on the heat, or if he was expecting mild conditions and it became cold on Shabbat, he may ask a non-Jew to turn the heat on.
With regard to air conditioning, too, there is room to allow asking a non-Jew to turn on the air conditioning if the heat is intolerable. Generally speaking, heat is not considered a potential health risk like the cold, and therefore, under normal circumstances, one may not ask a gentile to turn on an air conditioner or adjust an air conditioner on Shabbat. For example, if the air conditioning is running and one wants the temperature to be lowered another two degrees, he may not ask a gentile to adjust the thermostat. If, however, one did not expect hot conditions, or if a brief power outage disrupted the operation of the air conditioning, and the heat becomes insufferable, on may ask a gentile on Shabbat to turn on the air conditioner.
Of course, as mentioned above, heat and air conditioning should be turned on or set before Shabbat; the Halachot discussed here apply only if, for whatever reason, one had not set the unit before Shabbat.
Summary: Although it is generally forbidden to ask a gentile to perform on Shabbat any activity that is forbidden for a Jew on Shabbat, it is permissible to ask a gentile to turn on a light if a child is inconsolably distressed because of the dark. It is also permissible to ask a gentile to turn on the heat under cold conditions, or to turn on an air conditioner in situations of severe heat. Optimally, though, one should make a point of setting the heat or air conditioning before Shabbat.
