The question of what are essential elements is both straightforward and simple.
Whatever the story needs. (I never said it was useful. 0:)
But, you start with the story in hand. . . unless you don't. Perhaps you want to world-build. You find it fun. So you go off and world-build, like Tolkien, or like Hal Clement, who used to design a planet until it was no longer fun, then write down all the facts about it on index cards, then assemble them in a reasonable order of discovery, then write the novel to have the characters discover them in that order. It's useful to have some of that, because when the desire to world-build is less than the need to, you have a dull slog ahead of you. On the hand, if your desire to world-build greatly exceeds your desire to story, you have -- well, Tolkien, whose published work in his lifetime was never the stuff he really loved.
It's also hard to determine whether you have indeed reached the bare-bones minimum of a story. Plus world-building above and beyond that may water your story so that it sends off new shoots and blossoms. And its effects are wide-spread. It affects plot, and character, and local color, and realism. Henry James was wrong; it is not true that plot is character. Plot is character in setting. The story of a girl who wants to be a doctor is very different if she's an upper-class 19th century American, or a lower-class 21st century American, or the daughter of an impious man in a fantasy world where healing magic is strictly controlled by the temple of the healing goddess.
It's very likely that your idea comes with some world baggage attached. It should be examined as severely as everything else that suggests itself as an idea, but then exploited like everything else. If the great house has windows that would be vulnerabilities in any battle, for instance, it suggests a peaceable country. Perhaps steam-punk, or a magiteck equivalent. A dirigible would suggest that even more quickly. If a character is a princess, you have a monarchy, though how limited is still up in the air.
A vital thing to consider is whether it suggests how wild, weird, or wonderful your world is. Either way has enormous world-building implications. It is, for instance, a flaw in Robert E. Howard's stories that the pirate strongholds are nowhere on the major trade routes -- can't be, they don't go between large countries -- because he doesn't have significant magic transportation. And, for that matter, he has Euclidean geography. Most fantasy worlds do, of course. In theory. Tallying up distances is not always carried out perfectly, and supply logistics would be immensely difficult in many worlds. Then, if you decide not to have Euclidean geography, you have to work out how people move. Supply logistics still can be a difficult. Are the connections random or fixed? If they are random, your character can never land conveniently and obviously somewhere handy for them or your readers will know you put your finger on the scales.
Whenever a world has a major fantastical element, you must rigorously think through implications until you work out everything that will affect your character and his story. This can be a little less than obvious. If sailing ships magic up winds to fill their sails, does it affect the weather? If magical ability is inborn, how is it distributed among the population? Even if there is a magical caste, what happens with the little -- accidents, who do tend to be born? Not to mention love matches.
But where the element does not apply, or even in the aspects it does not affect of some things, you need to have realism, and world-building for it. The beautiful daughter of a rich merchant -- what did he make his money in? How desperate are he and her mother to marry her off to a nobleman? How desperate would a nobleman have to be to marry a merchant's daughter, sullied by trade? What sort of training would she have been given to make her more suitable as a noblewoman than to marry some other merchant? You may never need to touch on how the common children amuse themselves, or where they get the luxurious foods for the feast.
In both settings, the work helps you from the gravest danger -- to automatically reach for what is familiar to you in your own world, and slap it in the story. A strange world filled with wonders should always have another wonder.
Some people use questionnaires to think things through, which go through various aspects of the world to make sure you think of them. Me, I use extensive reading in primary source to build up a sense of what a society is like, and test against it. What techniques do you use?
part of bittercon