A coddled egg can be two things
(1) An egg, in its shell, that has been placed in boiling water for 1 minute, no more and no less. This is the original meaning of the word. Comes from the old verb, to coddle, which meant to boil or stew gently.
This is basically a very soft boiled egg. It can be used as an ingredient in cooking, for instance in a Caesar Salad dressing. It was highly recommended as nutritious for infants and invalids, before food writers changed their minds somewhat and told us a coddled egg would actually kill them. The danger is that a coddled egg is not cooked well enough to kill any salmonella that might be present. If you eat your fried eggs sunny-side up or over-easy, however, you're taking the same risk, even though not all the food writers seem to have connected the dots yet: currently, they keep focussing on a coddled egg as evil.
(2) An egg that has been broken open, emptied into a coddler, and cooked in a coddler.
A coddler is a small, heat-proof cup made of porcelain or pottery, with a metal top. The metal top usually screws on. You put the egg into the coddler, put the top on, and put it in a bath of boiling water that reaches about halfway up the little cup to cook the egg. Never use in a microwave, owing to the metal top, naturally, and never put the coddler directly onto a hot burner.
There's nothing to stop you actually coddling an egg, in the soft-boiled sense of the word, in an egg coddler, and that was probably the original intention, but probably most people now cook the eggs in them much firmer, making them very similar to a Shirred Egg.
Coddlers can be quite fancy, such as those made by Royal Worcester. There are many avid collectors of these cups, who would be scandalized to think that a "coddled egg" could be a plain old soft boiled egg that didn't use one of their treasured cups. Even though Royal Worcester says "dishwasher safe", they would also be scandalized at the thought of subjecting their treasures to a wash and rinse cycle. You could say -- they molly-coddle their coddlers.