How Things Might Have Been: Individuals, Kinds, and Essential Properties

How are we to distinguish between the essential and accidental properties of things such as individual people, cats, trees, and tables? Almost everyone agrees that such individuals could have been different, in certain respects, from the way that they actually are. But what are the respects in which they could not have been different: which of their properties are essential to their being the individuals that they are? And why?