What material is my damaged kitchen ceiling made of and how should I repair it?
Looks like basic plaster with a skim coat, which has released due to the moisture. The only question is whether you have wood lath base or gypsum panels. 

I'd use a putty knife to scrape away all the lose coating material. Skim with modern joint compound to achieve a flat finish. Prime and paint. The scope of the individual steps is too broad to lay out in an answer here. There's plenty of information to be found.

If the plaster itself has lost its integrity, you'll need to cut out the affected area, replace it with drywall (shimmed down flush), tape, prime, paint. It'll be a mess because all your insulation will fall out. Here's hoping you can avoid that. Plaster is pretty tough and may have tolerated the moisture.