I think I'm beginning to understand.
'Erase' in AP doesn't work as it does in other programs I've used.
In those programs, the erase toll automatically picks up pixels from the surrounding area and replaces the pixels in the erased area with them.
In AP, 'Erase' works by picking up pixels from another layer and replacing the erased area with them.
What the replacement looks like depends on what is in the other layer.
That seems to be what you, Alfred, have said and also what I understand is in the video that Cecil recommended.
However, I still have a problem.
I open the TIF file I want to modify. AP automatically adds a Background pixel layer.
I then go to 'Layer' and click on 'New Fill Layer'.
A fill layer is added at the top of the layer stack.
I then uncheck the fill layer (only way I can get back to the Background layer to work on it).
I click on the Erase Brush Tool.
Again, everything works - I can change opacity etc. and can drag and paint over the area I want erased.
The checkered area appears, but the color of the fill mask doesn't appear in the erased area - and that's it.
Nothing else happens, and as I said in my original post, I can't find any action/keystroke/mouse click that makes anything happen.
Obviously, either there's something about my hardware that AP doesn't like, or, much more likely, there's just some very basic things about working in AP that, lacking experience with Photoshop, that I just don't get - and in the few days left in my trial period - aren't going to get.
Too bad. I haven't been satisfied with the way erasing functions in my other main programs - although they work as advertised.
I had hoped that AP might be the solution.
BTW Alfred, you live in a magnificent country. Spent too short a time there 2005 - Plockton and a short while in Edinburgh. (My maternal grandmother's family were Campbells, traced back to Scotland.)
Thanks to both of you for trying to help.
Also, I did watch a number of videos on YouTube - could never get the good clean results shown in them - always artifacts left.