Peel, cube and boil the potatoes in salted water, as usual. (If you like Garlic-Mashed, simply add a few cloves to the water with the potatoes.)
When the potatoes are fork-tender, drain, reserving about an inch of water in the bottom of the pot with the potatoes. (You'll also need to catch some of the potato water in a cup or bowl at the sink, for later usage.)
A hand blender is needed for this part, but it's worth having one (and not just for potatoes!). Using olive oil instead of butter and/or milk, blend the potatoes with an up and down motion. Give them a stir, making sure you've gotten all the lumps. You can add a bit more olive oil or a bit of the cooking water to achieve the desired consistency.