Pubscreen is short for "public screen", it's simply a screen that can be used by any application (i.e. any application can open its window(s) on this screen). The most important pubscreen is "Workbench", i.e. the screen that is displayed when AmigaOS finished booting.
Not all screens are public (though non public screens are a lot less common than a decade ago). A programmer might decide to open a non public screen instead of a pubscreen, because that might make his job easier (he doesn't have to care about other windows that might be open on 'his' screen), it gives him the option of using faster screen drawing routines etc.
