Depends on the skin and the type of potato ...
Tough or unscrubbably dirty, it goes. If I keep the skin, I often score it first -- a bit of a pain sometimes, but it ensures the skin ends up in shreds in the mash rather than big chewy pieces. But there's a lot of flavour and nutrition in the skins, so I like to keep them when possible. Gives more of a rustic look, if that's wanted.
Also, some potatoes, esp. floury ones, just suck up the water, so the skin aids in preventing waterlogging and falling apart.
I sometimes cook the potatoes whole but usually halve them. I wouldn't quarter or cube them because then they can become waterlogged.
Similar to the Cook's Illustrated recipe, I always mash them first with a masher and then stir in butter and milk with a wooden spoon. I don't know where I picked up this method, but people often comment on how smooth and tasty my mashed potatoes are. I used a food mill once and couldn't tell the difference, though people who use ricers rave about the texture.
Depending on the rest of the main dish, I'll add a clove of garlic or two to cook with the potatoes, and mash them in, and/or various other roots: carrots, parsnips, turnips, celery root.
Daniel Boulud has a recipe in which he cooks potatoes and celery root with half-and-half combo of water and milk, then drains them, purees in a food processor, and adds just butter, no more liquid. I want to try this one day, also with just potatoes.