I am convinced that "classy" is the highest possible compliment one can give in contemporary English. I've been a freelance professional translator for many years now, and I've also found that it is one of the most difficult English words to translate into other languages. It simultaneously encompasses aspects of elegant, posh, stylish, sophisticated, polite, dignified, solicitous, considerate, conscientious, courteous, self-abnegating, high-class, fashionable, dashing, graceful, gracious, cool, slick, smooth, moral, ethical, thoughtful, proper, respectable, respectful, and rectitudinous, all without quite becoming a synonym for any one of them. In a way, it absorbs all the best possible connotations of each of those words, while leaving out any negative nuances. I can think of no word in English that I would rather be called than "classy," and while there are certainly more vulgar put-downs, it's hard to think of a more scathing insult than its antonym, "classless."
The internet yucksters over at Urban Dictionary similarly struggled to come up with a concise, cohesive definition. One of them tried to give a few and then kind of just gave up and deemed it, a deeper, more meaningful word for "cool". Given the status that "cool" has achieved in our language as one of the ultimate compliments, I guess that's as concise a definition for "classy" as any.