How can I use the same material at another slot and change it independently? I modelled something and want to select some specific faces for this object to give it another material. Works with applying. But now I have an existing material and don't know how to change it (e.g. make it darker) without changing it for all the other faces too.

When I click on the second material slot and want to change it and apply, the first slot changes too. How do I avoid that?
Click the New Material button - to add a new material based on your current one.
Then change the new material.