If I have a texture, is it then possible to generate a normal-map for this texture, so it can be used for bump-mapping?
Or how are normal maps usually made?
There's probably many ways to generate a Normal map, but like others said, you can do it from a Height Map, and 3d packages like XSI/3dsmax/Blender/any of them can output one for you as an image.
You can then output and RGB image with the Nvidia plugin for photoshop, an algorithm to convert it or you might be able to output it directly from those 3d packages with 3rd party plugins.
Be aware that in some case, you might need to invert channels (R, G or B) from the generated normal map.
Here's some resources link with examples and more complete explanation: