Generally speaking, cooking - including Bishul but also Afiah (baking) and other variants - is about "improving food through the application of heat". The key words include:
- Improving - e.g., if a food is fully cooked and then gets burnt/overcooked, that is not intrinsically Bishul
- Food - it gets expanded (as many other things) in applications relating to Shabbos, but as I understand it, Bishul normally applies primarily to edible substances
- Heat - there are some additional things that are similar (in particular, pickling or similar activity), but in most respects the key is heat.
How this gets applied varies quite a bit. In the case of solid food, simply adding heat that does not result in a change that materially improves the food does not count. That is why, within certain constraints, fully cooked (so no improvement) solid food can be heated up on Shabbos.
On the other hand, liquids are considered to "improve" simply by the addition of heat, even if they have been heated up before. As a result, if you have a kettle of cold water, you can't put it on a flame, even over a blech, on Shabbos.
How much cooking of a solid food counts as Bishul varies a bit. One key is Maachal Ben Drusai - 1/3 or 1/2 cooked. If it is cooked at least that amount then there are certain leniencies with respect to handling the partially cooked food on Shabbos. On the other hand, unless it is fully cooked, it can't be heated up on Shabbos if it was not already warm (and still cooking) before Shabbos.
Bishul Akum, as I understand it, uses largely the same rules for defining cooking, but the limitations are quite different from Shabbos. In general, a single step taken by a Jew - e.g., turn on the flame or even add to an existing fire - is sufficient to remove the prohibition of Bishul Akum.
Halachic cooking may, with the exception of liquids, where just raising the temperature is sufficient to count as Bishul, be a chemical change or a physical change. But normally we don't concern ourselves with the nature of the specific changes. Rather, it is "gets better" (cooks, bakes, etc.) or doesn't.