Peeling an apple isn't rocket science, but some methods are better than others. If you're looking to safely minimise the amount of flesh you lose while still getting sizeable chunks of peel you can use for other things, this method from the folks at Food52 is worth a shot.
You'll need a vegetable peeler and a knife for this one, and you cut the top and bottom of the apple off in small pieces so the apple is flat on both sides. This makes it easy to hold the apple safely to use the peeler on it, and lets you slide off flat, straight sections of peel that you can save for mulling spices, fruit jam, or even apple tea. You also end up with an easy-to-handle apple left behind that's simple to core or chop into chunks for baking or cooking.
If you're just going to eat the apple raw, this is a lot of work just to get the peel off (and who wants to lose the top and bottom?) but we can see it being useful if you're going to dice, slice or chop up the apple for cooking.
The Best Way to Peel an Apple [Food52]