I have downloaded a material off the internet and appended the material into my scene, but when I went to the node editor and tried to change the nodes, I can't.
They are simply locked:
Notice how the nodes are slightly transparent. How can I unlock the material?
There are two different ways of importing a data-block from another blendfile as said by the blender references manual's page:
Link creates a reference to the data in the source file such that changes made there will be reflected in the referencing file the next time it is reloaded.
Whereas Append makes a full copy of the data into your blend. You can make further edits to your local copy of the data, but changes in the external source file will not be reflected in the referencing file.
When you link a material, you'll not be able to edit that resource.
You can unlink & make the resource local (so you'll be able to edit the data-block) by clicking the little icon next to the material's name in the material tab:
