Interesting question, Lisa! 

The term 'elegance' when referring to a language can be described as graceful, refined, and sophisticated. It is often associated with a  well-spoken, well-crafted, and aesthetically pleasing manner of expressing yourself. This can include precise vocabulary, varied sentence structures, and a smooth flow of words. An elegant language can also convey a sense of sophistication and intelligence, making it appealing to both the speaker and the listener. 

In terms of 'register' it will be more formal than the way we speak to each other in a family setting or even at work. Depending on social situations, your language register will vary.

Hope this helps!