The Mysterious Material Drop-Down | Material linked to Object vs Data

Karim Joseph writes:

In this tutorial, we have a look at the material drop-down that we see all the time when we create materials in Blender. The one right next to the material’s name; material linked to Data or to Object.

What can we do with it?

Let’s find out by first looking at an example, and then with a step by step explanation on how to use it.
Hope it will be useful!