Any tips on how to do what I'm asking are welcome, but first of all I would like to ask for the correct terminology (because not knowing it prevents me from even Googling myself).
Maybe the title is not totally clear (because I don't know the right term), so let me explain in detail:
suppose you have a thicker plank or other dimension lumber, something like this
now let's say, you want to make the middle part more deep in a rectangular area, but not cut completely through the lumber, like this:
+---------------------+
| <------ untouched part
| +-----------------+ |
| | | |
| | <---------- deeper, flat, but not
| | | | cut through surface
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | |
| +-----------------+ |
| |
+---------------------+
One example use case, just to give an idea, would be building a table top from which the tools (like screw drivers, pencils, etc.) cannot roll down. (The actual use case I'm planning to do, is building a base for a power-strip, inspired by this answer.)
Questions
What is this process called? (I thought I remembered something like "embeveling", but that is not the correct word, because it means "chopping off the edge at a degree.) -- This is actually the most important.
What tools can be used to do this?
What is the procedure for doing this?