The sages banned the bread of non-Jews in order to prevent excessive socializing with them, which may lead to intermarriage (which could in turn lead to idol worship).14
Bread is considered "Jewish" if a Jew:
1. Turned on the oven; or
2. Threw a match into the oven; or
3. Placed the bread in the oven.
If the bread is made for commercial purposes it is permitted since it is not usually purchased in a social setting. The Shulchan Aruch allows this leniency only if Jewish bread is not available, whereas the Rema is lenient regarding commercial bread in all circumstances. (Note: this leniency obviously only applies if the bread is kosher and the only problem is that it was made by non-Jews).
14 Avodah Zarah 3Sb, 36b. Despite the fact that the prohibition stems from the concern that eating the bread of non-Jews may ultimately lead to idol worship, posekim rule that even where this concern is not relevant, the prohibition still stands. Thus, for example, eating bread cooked by a Muslim is still prohibited despite the fact that Islam does not constitute idol worship. (Peri Megadim Siftei Da'at Y.D. 112/2).
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