As said, I am not Master Egg-Poacher. Nevertheless, here is a tutorial–there are many different techniques, but the general idea of cooking the egg in gently simmering water is the same.
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1. Yes, you do need to use fresh eggs. You don’t need to steal from your neighbor’s coop (although ask if you have a neighbor with a coop), just know that those eggs that have been sitting in your fridge for weeks are not going to do that well.
2. Bring a pot of water to boil on the stove. You need at least 3 inches and a skillet or pot that will allow the egg to move some.
3. While waiting for the water to boil, prepare yourself. Have your slotted spoon ready and crack an egg into a bowl.
4. When the water just begins to boil, throw some salt in and a few tablespoons of vinegar. This should stop the water from boiling. Turn the heat down to a simmer.
5. Using your spoon, stir the water so you can get a gentle whirlpool.
6. Slide the egg into the center of the whirlpool.
7. The egg should coagulate almost immediately. If any of the less-than-desireable things should occur, like the white sticking to the bottom or some stringyness, just laugh it off.
8. Keep the egg in there at a simmer; you don’t want the water to boil.
9. I like to cook mine for about 2 minutes, although the timing varies from person-to-person depending on how runny you like your egg.
10. Remove the egg with your slotted spoon (so it drains water).
11. Poached eggs ooze, so it’s really nice to have them with something that can sop up the goodness: on a piece of toast or croissant, or, if you don’t do the carb-thing, on a nice bed of spinach or sprouts.
