Different Ways of Cooking Eggs
Scrambled Eggs (Hard Scrambled)
Scrambled eggs involve mixing/whisking the egg yolk and whites together, mixing together the yolks and whites of however many eggs you're cooking with. No mercy. Hard scrambled eggs are fully cooked scrambled eggs and typically what you'll get if you order scrambled eggs at a cafe, restaurant, etc. The problem is they risk being "too done," dry, etc.
Scrambled Eggs (Soft Scrambled)
These are scrambled eggs with a shorter cooking time, leading to scrambled eggs that end up a bit wet. Since eggs cook so quickly, that means even shorter than usual - you'll need to keep a close eye on them. These are better eggs: better texture, better taste, better everything. Whisk the eggs beforehand in a bowl. Heat the oiled pan to no more than medium and have your spatula at the ready. Mix and fold the eggs as they cook, but don't let them spread out too much in the pan or they're easier to overcook.
For another perspective on how to scramble eggs, see this article.
With omelets (and other variations like frittatas), you mix up the egg yolks and whites beforehand, but you don't turn/mix/bother the eggs once you get them into the pan. Then, of course, you add your other ingredients/toppings once the omelet eggs have cooked sufficiently.
Poached eggs are eggs you boil but without the shell. Add the cracked eggs to a small bowl. Bring water to a light boil in a big pot. Use a long spoon to stir the water and create a whirlpool effect. Now add the eggs - the whirlpool will keep the eggs together in the water while they cook. You'll want to cook the eggs for about 2 or 3 minutes and then take out with a slotted spoon.
Fried - Sunny Side Up
Sunny side up eggs are those eggs with bright yolks. Crack the egg into an oiled or greased pan and let if fry up to the point the sides brown. Don't flip the eggs. The yolk will be runny and open.
Fried - Over Easy, Over Medium, or Over Hard
Over easy eggs repeat the process above but now you're going to flip your eggs then cook them briefly. The yolk will be runny but now covered. Over medium eggs are cooked longer once they are flipped. Over hard eggs are cooked even longer (all the way) once they are flipped.
Hard boiled eggs are boiled in water with the shells on (you remove them after cooking and before eating). Put enough water in a pot to cover the eggs you're hard-boiling with a few inches of water. Let the pot of water come to a boil and then add the eggs in. Cook for somewhere between 10 and 12 minutes. It can sometimes be difficult to remove the shells. If you want to make that part easier, put the eggs in very cold water when they are done and then carefully knock them against a counter or other surface.
Soft-boiled eggs go through the same process as hard-boiled eggs, but with less cooking. Now you want to cook the eggs for more like 5 or 6 minutes (about half the amount of time). You will end up with a white that is similarly cooked but a yolk that is softer or runnier.