What could a thing be in your eye that is like a piece of fuzz and moves with your eye movement?

These are called "floaters".
It is possible that it is a small - very small - piece of detached retina or more commonly, structural debris from the vitreous humour. As we age, the vitreous humour, a jelly like sac that fills our eye cavities, can begin to detach and condense from the sides of the eye and the retina. The protein threads that hold this together can become lose and float through the eye as the vitreous humour becomes more liquid. Sometimes, as the vitreous detaches, it can rip or tear the retina along with it. This is an emergency!

If you notice you have floaters, you should see an eye doctor immediately to rule out retinal detachment. Fortunately, most people's floaters develop without harming the retina. In fact, most people notice floaters from time to time and they are mostly normal and benign. Once you've seen a doctor to make sure everything is okay, make sure to see them again if you notice your floaters worsening, or if they are accompanied by flashes of light.Floaters may be annoying and distracting. If that is the case, rest assured that you will learn to ignore them with time. However, sometimes floaters can be debillating. Some vitreous retina surgeons will perform a vitrectomy to remove floaters if they feel it is warranted. In vitrectomy surgery, the vitreous humour is sucked out of the eyeball and replaced with a saline solution. The operation is done on an outpatient basis, and most patients sight returns to normal within a matter of weeks. Although the complications of vitrectomy are treatable, they include cataract, retinal detachment and rarely, infection. Vitrectomy is an invasive surgery that should only be considered if you have floaters that significantly interfere with your vision and/or your quality of life.