Sunny Side Up -- Not flipped, unbroken yolk. The top of the egg is just barely set.
Basted -- Sunny Side Up, hot fat spooned over until the white surrounding the yolk is opaque.
Over Easy -- Flipped, unbroken yolk, yolk runny.
Over Medium / Well -- Flipped, unbroken yolk, yolk cooked to have a firm but wet-appearing center.
Over Hard -- Flipped, broken, fully-cooked yolk.
Broken / Lightly Scrambled -- Broken in pan and gently stirred while cooking - yolk and whites should not be mixed entirely.
Scrambled Eggs -- Made in many different ways. Generally the eggs are mixed in a bowl before being put into the pan, and often stirred while cooking. Some recipes add fat to the eggs in the form of milk, cream, butter, or oil. A distinction can be made between Wet/Loose or Dry, which refers to the degree of doneness.
Filled Omelette -- Eggs mixed before cooking, possibly with added fat as in Scrambled Eggs. Cooked in fat in a saute pan; when set but the interior still wet, previously-cooked fillings (cheese, onions, mushrooms, peppers, tomatoes...) are added, and the eggs folded over into a half-moon shape.
Spanish Omelette / Western Omelette -- Same as filled, but the egg mixture is poured over the fillings in a hot pan and cooked, thus incorporating the fillings into the egg.
Fluffy Omelette -- Whites and yolks beaten separately. Yolks are gently folded into the whites without breaking the structure of the whites. Optional toppings are added. Cooked slowly in a pan, or baked (an electric frying pan with a lid works well for this preparation).
Cooked in shell in water for a timed period. Some people will refer to degree of doneness by cooking time, i.e., a "3-minute egg" is soft-boiled with some runny white around the yolk. Some recipes call for eggs to be added to boiling water, others to be started in cold water. In the cold-water start, the pot may be left on the heat or removed when the water reaches a boil. The eggs may be shocked in ice water when removed.
Soft -- Yolk runny, potentially with slight unset white around the yolk.
Medium -- White completely set, yolk firm but with a dark, wet appearance.
Hard -- Yolk completely set and pale yellow.
Egg cooked, out of shell, in water, stock, or other liquid -- excluding fats or oils -- at a temperature in the range of 160-180˚F (70-82˚C). There are possible degrees of doneness, however the typical poached egg has a runny but warm and thickened yolk and fully-set white.
An egg that has been very lightly cooked (poached eggs are sometimes considered coddled). This can either be accomplished with an egg coddler or cooking an egg in its shell with water that is slightly below boiling point.
An egg cooked in an oven at the low-mid 300's˚F (~160˚C), contained in a ramekin, until the whites are just set and the yolk is runny but thickened. Often butter or another fat is placed on top before cooking.
Very similar to shirred eggs, but the ramekin is covered with some aluminum foil and put in a steamer for 7-15 minutes.
In a basket:
Eggs fried in a hole made in a slice of bread