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Although a non-Jew may certainly do melakha on Shabbat, the Sages prohibited a Jew from asking a non-Jew to do melakha for the Jew on Shabbat. They also prohibited a Jew from benefiting from melakha done by a non-Jew on behalf of a Jew.
For example, if the lights in a Jewish home went out, and a non-Jewish neighbor came and turned them on for them, the home’s residents may not benefit from the light. However, if the neighbors did the melakha for themselves, say, to read a book, Jews may benefit from the light. Moreover, even if non-Jews intended to turn on the light for both themselves and the Jews, they may benefit from it, since the non-Jew needed to turn on the light for their own benefit in any case.
Therefore, if the lights have gone out in a Jewish home, they are allowed to invite a non-Jewish neighbor to come over for a snack. When the neighbor arrives and sees the house is dark, the hope is they will realize the lights need to be turned on. Since the neighbor is turning them on for their own benefit to see the food being served, the Jews may benefit from the light as well.
The Sages permitted asking a non-Jew to do a rabbinically forbidden melakha for the sake of a mitzva or in case of great need. For example, if the lights went out on Friday night before dinner, a Jew may ask a non-Jew to turn them on with a shinui (such as using the back of his hand). Three factors combine to allow this: the prohibition upon asking a non-Jew to do melakha is rabbinic, doing melakha with a shinui is rabbinic, and the purpose here is to enable the fulfillment of a mitzva (enjoying the Shabbat meal).
When there is an especially great need, the Sages permitted asking a non-Jew to do a melakha prohibited by Torah law. Examples are buying land in Israel, saving someone from a very serious financial loss, and caring for someone sick (even if the illness is not life-threatening). However, these leniencies are limited to very rare cases, and may not be incorporated into a person’s regular Shabbat routine. The only exception is when a non-Jew is caring for a sick person who needs constant care. (If someone is dangerously ill, Jews too must do whatever melakha is necessary to keep them alive, as danger to life supersedes Shabbat.)
