How do you run Drupal core-patches and is there a single, general way? When I run contrib-module patches I download the patch file, put it in the modules folder, navigate to there with the Linux-bash and then do git apply patch patch_file_name.

I never ran a core-patch though and I believe it's a bit different as there isn't necessarily a single module to act upon.

So, is there a one single way to run Drupal core patches and how it should be done without damaging anything? Maybe just put it in the site root folder and run it?

I ask this here so people who write such a question in Google could find this info easily.
Drush make is a method to maintain patched releases of contrib and core. It is the only organized, non-custom, method I know of for patch management. The makefile is self-documenting. By "self-documenting", I mean that the makefile contains a list of modules, their versions, and URLs of applied patches.

Create a makefile of an existing Drupal site with:

drush generate-makefile mysite.make


Note, as far as I know drush generate-makefile will not detect applied patches. The example makefile demonstrates how to reference patches in your makefile.