There's an entire chapter of Shulchan Aruch (Yore Dea 112) — sixteen subsections plus all the commentaries and later works — devoted to this, so obviously answers here, including this one, can't do the topic justice. But the Star-K and the OU summarize the main points, qq.v., and I'll briefly summarize the summaries:
- It's forbidden unless a Jew was involved in the baking.
- An exception is made for commercially baked goods, though for S'faradim only in case of need. (Many, even Ashk'nazim, don't rely on the exception at all, or at least not between Rosh Hashana and Yom Kipur inclusive.)
There's also a prohibition on eating bread that contains meat or milk in it unless it is markedly meaty or dairy. Again, there's a whole chapter of Shulchan Aruch (Yore Dea 97) on this, and the details are too many for this answer.