If a line or pattern is etched into a surface, it is cut into the surface by means of acid or a sharp tool. You can also say that a surface is etched with a line or pattern.

If you say that feelings are etched on someone's face, you mean that the person is very strongly affected by them, and you can see this in their appearance.

If something is etched on your memory, you remember it very clearly, usually because it has some special importance for you.