Even more particularly, when we talk about physical coordination, we are talking about several interacting systems. Humans have a complicated nervous system. The peripheral nervous system (PNS) is spread around the body. It gathers information about the body and its environment, and delivers that information to the central nervous system (CNS). It also takes information and commands from the CNS and delivers it to the body to organize movement, etc.
When it comes to physical motion, we can (for simplicity's sake) break the process of coordination up into 2 or 3 main systems. The first two systems have to do with where the body is and how its moving. There is a dedicated part of the brain and nervous system that tracks where all the parts of your body are at any given time. This type of sense is called proprioception. It is how you know roughly where your hands and feet are, even with your eyes closed. (Example: close your eyes, and try to bring your hands together. Then try to swing your hands past each other without touching. It shouldn't be too difficult, because you know/can sense where your hands are in relation to each other. This is proprioception.) In people who have brain damage in the region of the brain that controls proprioception, they literally cannot walk or move without paying conscious attention to where their arms and legs are. Their body becomes more like a robot they have to consciously control by thinking commands at it.
You also have a vestibular system. This is mainly a part of your inner ear, and it is how your body senses movement. You've got a set of semicircular tubes in each of your inner ears. These are filled with a thickish fluid, and lined with small hairs. When you change how you're moving, the fluid in these tubes wants to keep moving the way you were before. When the fluid continues to move in a straight line (thanks, inertia!) or as close to a straight line as it can in those curved tubes, it presses against the hairs. When these hairs bend, it sends signals to your brain telling it that you're moving differently. You also have small stoens in your ears that serve a similar function, to know when you are moving in a strait line, and how fast.
When a kid spins in a circle, they feel dizzy for a moment as the fluid in their ears presses against the hairs in those tubes. After a little bit, the fluid catches up with the spinning body, and stops pressing on the hairs. Then the kid gets tired, and stops. Now, the tubes in their head have stopped moving, but the fluid keeps going around and around. This again presses on the hairs, which again tell the brain that the body must be moving. The brain interprets this motion as the world "spinning." So the vestibular system is all about balance and movement. (When I went deaf, I literally couldn't stand up for days, because the vestibular system in my right ear had been destroyed. Without it, I couldn't accurately sense how my body was moving, and therefore couldn't walk properly.)
Finally, there is the part of the nervous system that senses the outer world through sight, sound, etc.
What does all this have to do with dancing? Well, dancing requires very complex coordination. To move, your body must "know" where the limbs are in relation to each other, so they can be brought together to make the correct shapes and forms. They must receive accurate and precise information on how to move in order to create that form. You must know where the floor is, where the wall is, how long the drop will take to reach the floor after leaping, etc. Any small problem with any of these systems can make precise coordination impossible. If you don't accurately sense where the floor is? You'll bring your weight down at the wrong time and stumble. Bad dancer. If your vestibular system isn't working perfectly, and you can't sense how fast you're moving? You'll bring your weight down at the wrong time and stumble. Bad dancer. If your proprioceptive abilities aren't finely tuned? You won't bring your foot to the right place, because you won't be certain where your foot actually is. You bring your weight down at the wrong time and stumble. Bad dancer.
That said, the brain is fairly plastic. The nervous system can grow and improve. Practice makes perfect, as they say, or at the least makes one better. To be sure, there is a level of innate talent, an inborn natural genetic ability to synthesize all these senses to be coordinated. But skills can be grown, as well. Someone with naturally bad coordination can work and practice and improve to be better than someone with a talent for coordination who never practices.