dictionary.com defines elegance as
1. tastefully fine or luxurious in dress, style, design, etc.: elegant furnishings.

2. gracefully refined and dignified, as in tastes, habits, or literary style: an elegant young gentleman; an elegant prosodist.

3. graceful in form or movement: an elegant wave of the hand.

4. appropriate to refined taste: a man devoted to elegant pursuits.

5. excellent; fine; superior: an absolutely elegant wine.

Before a recent edit, the Wikipedia entry for elegance included “the concept is vaguely and arbitrarily understood” then went on, “Nonetheless, essential components of the concept include simplicity and consistency of design, focusing on the essential features of an object. In art of any kind one might also require dignified grace, or restrained beauty of style. Visual stimuli are frequently considered elegant if a small number of colors and stimuli are used, emphasizing the remainder.”

Some favorite quotations that give some interesting angles on the subject (found on thinkexist.com):

Elegance is refusal.—Coco Chanel

The only real elegance is in the mind; if you've got that, the rest really comes from it.—Diana Vreeland

Isn't elegance forgetting what one is wearing?—Yves Saint Laurent

To me, elegance is not to pass unnoticed but to get to the very soul of what one is.—Christian Lacroix