I started social dancing two and a half years ago, and it's difficult to explain what it's all about. The term "dance" encompasses a huge range of activities, and even if you give examples of specific styles (e.g., waltz, swing, salsa, quickstep), most people have no idea what it actually involves or why it's so enjoyable.
The core of social dancing is communication.
It's not "listening to the music", it's not "moving your body", and it's certainly not "memorize these steps and imitate what you saw on TV". The fundamental part is non-verbal communication between two people.
The best way illustrate this is one of the simplest partner dancing exercises: "the shopping cart".
Two people face each other and hold hands, slightly wider than hip-width apart. One of them will be the "leader", and will begin the movements. The other person is the "follower", and will respond to those movements. The goal of the follower is to stay the same distance away from the leader. The goal of the leader is to move clearly enough that the follower can react accordingly.
For example, if the leader steps forwards, the follower should step backwards. If the leader takes a small step backwards, the follower should take a small step forwards. Simple enough, right?
It sounds simple, but there's a lot of complexities that we can add. What if the leader takes a big step forwards, then a small step, then a big step backwards? What if the leader steps to the side? What if leader takes a slow step forwards, then two quick backwards steps, then a slow step to the side? Etc. There's no pre-determined choreography here -- the leader makes it up on the spot. The only way the follower knows what to do is by reacting to the subtle pressure of their hands.
The name "shopping cart" comes from a helpful visualization -- the follower pretends to be a shopping cart. Shopping carts have simple physics: the metal frame doesn't bend, and as long as the wheels aren't broken, it moves exactly where you push it. If you keep your hands out and walk forwards, the cart moves along. If you stop moving forwards and step backwards, the cart follows. If you push with one hand and pull with the other, the cart turns in one direction. And if you push the cart and let go, the cart keeps on moving until friction slows it to a stop.
Bonus round: do the exercise with the follower's eyes closed. That's when you really test the dance connection between the two people!