I recently moved and noticed this crack on the wall. When I tapped it lightly with a blunt knife it broke off. I've never plastered before but I assume I will need to re-plaster this section. Any advice would be much appreciated.
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.
That looks like poorly-adhering paint to me, possibly on top of a crack in the plaster. You'll need to remove all the loose paint you can find. Then, feather the sharp edges of paint with a thin layer of plaster, prime, and paint the wall.