How do I repaint pealing painted cabinets?

You probably need a primer plus a good quality latex enamel paint. If the cabinet surface is laminate of varnished, you will probably need to use a de-glosser.

If your cabinets were originally painted with oil based paint and you try painting over it with latex, it will not stick - peels right off. Also it is important to thoroughly clean your cabinets before you begin regardless of the paint you plan to use. May I suggest you take down one of your cabinet doors and take it to a profession paint store (not a big box store) and ask them to test the paint on it to determine what you have so that you can get guidance from them as to what product you need to complete the project?

This problem can be caused by some layers of paint being oil and some being water based paint. Or, the cabinets had an oil finish or grease before anyone painted them. The best thing would be to sand down to bare wood and then use a great primer like Zinnser.

You just need to lightly sand to roughen the surface for paint. Prime with Kilz then paint.

How to Paint Cabinets

Make sure they are clean and dry. Remove the doors and hardware. Mark the doors and cabinets with tape where they go. Lightly sand the doors and cabinets to remove any gloss and roughen the surface for paint with 220-grit sandpaper. Use a tack cloth or damp rag to remove dust after sanding. Prime with a stain blocking primer like KILZ. Acrylic, or water-base, paints are low-fume and clean up easily with water. Alkyd, or oil-base, paints require good ventilation because the paint contains solvents that can irritate your lungs and make you feel sick. Alkyd options require mineral spirits for cleanup, but they provide a hard, durable paint finish. Whichever you use, buy the best-quality paint you can afford for a lasting kitchen cabinet finish. Seal with at least three coats with a water based polyurethane. Use a small foam roller and foam brush for a smooth finish.

I would think at this point these cabinets need to striped or sanded back to the raw wood - there is an issue between layers and best to remove everything and start over. Once they are back to raw wood - I suggest XIM Bonding primer, apply two coats and then you can paint on a good quality paint like General Finishes Milk Paints they also have a great top coat. https://shop.artisticpaintingstudio.com/