If you etch something you are cutting into it. You can etch "I ♥ U" into a tree trunk with a pocket knife or etch your initials in your desk with a key (if the teacher's not looking).
However and whatever you etch, it is generally considered a deep and lasting mark. The figurative use of etch is just as common as its literal sense. A particularly powerful memory, image or emotion is said to "etch itself" into your mind, meaning, like that piece of graffiti you etched into the tree trunk, it won't be likely to be erased anytime soon. Incise and carve are often interchangeable with etch.