Android background mode - how to understand?

Question

Android

What is background mode?
There are applications that are not visible in the list of open ones - this is the background mode, there are no questions here.
And those that are visible in the list, but work is currently going on with only one (not with them) - what are they?
It's just that there is this very "autorun and autoclose", it says - prohibited applications do not start in the background or at startup (this is clear, although not true), and, (!) Close in sleep mode when the screen is turned off.
Since the application is not already running, it seems to be about applications in the task list, but not in the foreground.
And I need something that I don’t need to not work in a normal background, but if I launched it, it worked, when switching it was exactly in the same state in which I left it.
Am I understanding correctly or not?