Generally speaking, Halacha forbids instructing a gentile to perform Melacha (activity forbidden on Shabbat) on a Jew's behalf on Shabbat. Moreover, as noted by the Mishna Berura (commentary to the Shulhan Aruch by Rabbi Yisrael Kagan, 1839-1933) in Siman 276 (11), if a gentile voluntarily offers to perform a Melacha on a Jew's behalf, the Jew must object and instruct the gentile to desist. Even though the gentile acts voluntarily, and not at the Jew's behest, the Jew must nevertheless voice his objection since the gentile seeks to perform a Melacha on his behalf. (Needless to say, one need not stop a gentile from performing Melacha on Shabbat for his own purposes.) This applies even if the Jew has no intention of deriving any benefit from the action the gentile seeks to perform. If the gentile acts in order to do a service to the Jew, the Jew must object.
One common example of this Halacha arises when a non-Jewish housekeeper volunteers to turn on a light in a dark room for her Jewish employer. Not only does Halacha forbid instructing or asking her to turn on the light, it also requires the Jew to tell her not to turn on the light if she offers to do so. Similarly, if the gardener arrives on Shabbat to work on the lawn, one must instruct him not to do so. (Of course, one should clarify from the outset when hiring a gardener that no gardening work should be done on Shabbat or holidays.)
If a person sits in a dimly-lit room, where there is sufficient light for whichever activity he wishes to perform in that room (e.g. reading or eating), he may indirectly ask a gentile to turn on a light by indicating his desire for additional illumination. Thus, for example, if the lights are turned off but the sunlight from the windows provides enough light to read, one who wishes some additional light may say to his gentile housekeeper, "This room could use some more light." Although he may not explicitly request that she turn on the light, he may, in this situation, express his request indirectly. This is the ruling of several authorities, including the Hida (Rav Haim Yosef David Azulai, 1724-1806) in his Birke Yosef (307:6), the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) in his Rav Pe'alim (Orah Haim 83), and Hacham Ovadia Yosef in Halichot Olam (vol. 3, p. 179).
If the room is dark, however, such that one cannot perform the desired activity in the room, then he may not ask a gentile even indirectly to turn on a light. Indirect requests are allowed only in cases where there is sufficient light and one seeks additional illumination.
Summary: One may not ask a gentile to perform Melacha on his (the Jew's) behalf on Shabbat, and if a gentile offers to do Melacha for a Jew, the Jew must object. If a dimly-lit room has sufficient light but one desires additional light, he may indirectly express a request to a gentile to turn on a light, but he may not ask directly.
