Buddha-nature is something more easily understood intuitively, rather than by pushing it through the sausage grinder of your mind. Even so, I will discuss what it is, and hopefully it will help to make another begin to understand it.
Buddha-nature is, at its essence, self-nature. It is the natural state of the mind without intellectual or ego entanglements. It entails the understanding that all beings are all interconnected in this way.
It is by wresting control of your mind from your ego and its many invasive thoughts that Zen Buddhists seek satori, or enlightenment. In fact, satori is considered to be the moment of self-realization. It is the precice moment when you truly realize your Buddha-nature. As such, do not think that this writeup is going to be able to tell you precisely what it is.
As Phillip Kapleau Roshi defined it, Buddha-nature is, "A concrete expression for the substratum of perfection, of completeness, intrinsic to both sentient and insentient life."
The monk asked Chao-Chou (Joshu in Japanese), "Does a dog have Buddha-nature?"
Chao-Chou replied, "Mu".
Although the word "mu" meant "no" or "does not have" in Chinese, the answer is far deeper than that. Do not look on the surface for the answer, because it is not there. Obviously a dog has Buddha-nature, as it is a being, and all beings have Buddha-nature. This is the first Koan given to students in most sects of modern Zen, and once you understand it, you will have a fair understanding of Buddha-nature.