My mom used to love making deviled eggs. I've never cared for them. You take hard boiled eggs, cut them in half (after peeling), scoop out the yolk and mix it with certain ingredients (mayo, mustard and some other stuff). Then you scoop the yolk mixture back into the eggs and sprinkle with paprika.
There are different deviled eggs. Almost all start from cutting hard-boiled eggs from the pointy end to the blunt end, then removing the hard yolk. As Bip said, mixing the yolk with "other stuff" and then replacing the mixture in the white of the egg makes a "deviled egg." I like to mix the mashed yolk with Dijon mustard, a bit of mayonnaise, some drained pickle relish, a bit of salt and white pepper, and capers. These deviled eggs make a nice alternative to salad, and we enjoy them with sandwiches.
The fresher the egg the more difficult to peel. I boil 'til done and then plunge them into cold water. Helps a bit but still kinda tricky with brand new out of the bum eggs
Deviled eggs are hard boiled eggs cut in half . Scoop out the center and add things that you like. I usually chop some onion very fine and then add some sour cream and mayonnaise so it is kind of pasty. Then you scoop it back into the half of the eggs. Some recipes call for chopped broccoli, bacon.
I always drain them when they are done, add cold tap water, plus one of those reusable ice packs. Ice cubes work well too. I let them cool like that, and they nearly always peel easily that way. Though, like others observed, if they are too fresh they tend not to peel well. I use my freshest eggs for cooking, but any time I have extras, I use them for boiled eggs. I prefer fresh eggs also. If you know about when they were laid (like having your own hens), it won't hurt to let them set a few days in the fridge. Store eggs, you just never know.
That is what we do now, we store them in the fridge a bit for boiled eggs. They do peel better. That Russian trick with fresh eggs works quite well.