Probably one of the most abused preparations in the history of modern food, mashed potatoes have been around a long time, and only recently has anyone given some thought to proper preparation. That is, to serve potatoes that are not gluey, stodgy or too lumpy. (some people like lumps, or "texture") With properly cooked potatoes, though, the lumps are not necessary for texture.
In my opinion, the following are essential for good mashed potatoes.
1.) Choice of potatoes: You want floury potatoes, not waxy. Waxy potatoes are more suited to potato salad. Some cooks like to combine the two types - not me.
2.) Cut the potatoes into 1 cm slices, peeled or unpeeled, your choice. I find that unpeeled taste better. Just cut off any blemishes.
3.) Put the potatoes into heavily salted water. (20 g/l)
4.) Bring the temperature up to between 80⁰ and 85⁰ C. Hold at that for 20 minutes. You need a thermometer and have to stay nearby for this, (unless you have a sous vide heater / circulator) but it is worth it.
5.) Bring to a rolling boil, turn down the heat and simmer until the potatoes are just cooked, and drain.
6.) Put through a ricer or use a potato masher after first removing the peels if you wish. Add butter (a certain famous French cook uses the same amount of butter as potatoes by weight. That is possibly too much for everyday potatoes, but should be tried at least once.) Depending on the amount of butter you used, you may want to add a little milk or cream until the spuds are the consistency you want.
The idea behind the steady temperature of max 85⁰ C for 20 minutes is to keep the cell walls as intact as possible. When the cell walls burst the released starch absorbs water and that is what causes the gummy texture so often found. Whatever you do, never put the potatoes in a food processor or other mechanical device. Just mash them by hand until you have the texture and consistency required. Adjust the seasoning, I prefer a grating of nutmeg, black pepper and salt. Try this method, you will get the best results ever.
I did not mention any of the potato varieties, but think probably the most used is your basic russet potato. There are thousands.
Thanks to potato man for the image.