This is done with a special roller and common plaster/mud. It's just sort of hackle-dackle spackled on there, really amazingly cheap.
The purpose is to roughen the surface to conceal lack of parallelness in the ceiling panels, drywall taping lines, hole patches, and other surface defects.
What you have is a drywall ceiling that has had popcorn sprayed over it. You can paint over it but you'll have to repair any gouges made from scraping off the popcorn and then prime it. To remove the popcorn, spray water on it, let it soak in for a few minutes and then scrape it off with a mud knife. The larger the mud knife, the less you'll damage the ...
In the UK that type of finish would almost certainly be produced with Artex - a product that is still available in the UK, though I don't know about other countries.
My father used to produce the edge feature simply by running a paint brush along the edge after the main texturing was completed.