How to repair flaked plaster surface
I'd tap the rest of the wall and see really how much you need to repair. From the photo it looks as though you will have another inch or two upwards and at least another inch going down. This is a really easy fix though. You can get a tub of mud (plaster patch) at any home improvement, hardware or even your local department store and even large grocery stores carry it. Get a putty knife and it would help if you got a 6" or 8" drywall knife to make a smoother coat (saves sanding time). Then lookup how to patch drywall and/or drywall finishing on YouTube. There's a ton of great short videos that will show you better than we can tell you.