Beautiful dancing is dancing with the music.
The most unfortunate thing about this statement (like your questions) is that it is very deceptive. It appears simple. In some ways it is simple, and in some ways it isn't. The question you're asking requires us, as those answering it, to provide several definitions, or operate under the same assumptions and definitions of the terms in question: (i) beautiful, and (ii) dancing.
It is said that 'beauty is in the eye of the beholder.' This guarantees you a myriad of answers because every person here has his/her own understanding of beauty and of dance. I'm interested in what people will say, especially because your question isn't talking about professional dancers but rather social dancers. Here, again, a trick emerges: we must all operate under the assumption that social dancing is not any less demanding or in any way unequal to competitive. This is quite unfair as the two serve entirely different purposes in their existences ('competitive' is for a mastery of one's body for the sake of doing something significantly better than others [on the same floor], and 'social' is for socializing and falls into the realm of a hobbies more than anything else). For the sake of argument, let us assume that both are equal, and the dancers in question share a deeper understanding of dance than what John Q. Public (to say John Q Dancer implies a competitive spirit

) would have.
In any event, in answer to your question, my definition of 'beautiful dancing' is dancing with music. Notice, I say
with and not
to because there is a difference (to me) between stepping while noise is playing in the background and letting said noise be filtered, turn into music, infiltrate one's spirit and body and be moved by it. As dancers, music is the one thing that must move us; that is a non-negotiable rule. Without music dance would not exist, and so first and foremost it is imperative that the dancers on the floor move with the music. But even here, I must begin to split hairs. When we say 'dancing,' what do we mean? I see dance as something entirely different than let's say fascination sees it, or CANI sees it (I assume I do anyway). Dancing entails in itself a lot of little details. One cannot dance without proper posture, one cannot dance without knowing even the basest of steps, or even the most rudimentary principles of leading or moving in general. To me, dancing encompasses being able to carry out the steps flawlessly, with the proper technique, while maintaining the natural posture designated to the style of dance being performed.
With that defined, we move on to what one considers beautiful. Generally speaking, beautiful means pleasant to look at, enjoyable even. When a step is carried out with the correct technique, while the posture is maintained, and the lines and timing and rhythm are there, then it is correct... for a step (or dance) to be beautiful, all of the aforementioned characteristics (criteria) must be met, and the dancers must -- and this is non-negotiable for me -- put themselves into it, and by that I mean one's emotions. Without a person's emotions put into the steps, into the dance, all one sees is technically sound dancing. And that is not enough for it to be beautiful. It is enough for it to be correct. However, 'correct' and 'beautiful' are two different things, and on this point I doubt anyone would argue.
So, then: beautiful dancing is the carrying out of the designated and allowed steps with the proper posture, technique and styling in time with the music and infused with the personal and unique characterization and understanding of the steps and the dance achieved by the dancers.
And in looking at them, I do not care if they are professional or social dancers. A dancer, of any style, any caliber, should have enough
self respect to want his/her dancing to be beautiful, and should understand what makes dancing beautiful (in accordance with his/her own definitions of these terms). Dancing leads to self discovery, and self discovery prompts the questioning of all these things, which in turn influence how one sees the world, the self, and then one's dancing... and all is full circle.