Elegant cars, artwork, curtains, furniture, buildings and so on are fancy but simple; they're designed with care and skill. Here's the Eiffel Tower, an elegant structure. Its design is clean, shapely, and simple: that's elegance.
You can have elegant posture: you stand tall and proud. You can have elegant handwriting: your writing is beautiful and well-controlled. And you can play the piano with elegance, or dance with elegance: with grace, skill, and simplicity.
Lastly, we can call something elegant even if it's not visually graceful or artistic. You might come up with an elegant solution to a problem: a solution that's simple, smart, thoughtful, and creative.