As Henslin points out, race is a term which refers to a set of inherited physical characteristics. Race is, therefore, determined by biology, not by anything else. Skin color, for instance, is considered a racial characteristic.
In contrast, ethnicity refers to the shared cultural characteristics of a group of people. Unlike racial characteristics, these are not based on biology and are not inherited from our parents. An example of an ethnic characteristic is a shared language or a religious belief which unites a group of people.
Society's attempts to classify people into different racial groups shows just how complex this concept really is. As Barbara J. Fields argues, race is, in fact, ideologically and historically constructed. Take, for example, the idea of the black race. People in this racial group share only one common biological feature, their skin color, but do not share any other biological characteristics. Moreover, the specific tone of skin color can vary significantly.
To try and overcome this issue, we can narrow down this racial term, an example being "African American," but still, we face the same problems. (For the full article, see the reference link provided.)
In addition, as history shows us, race is often tangled up with cultural beliefs. Colonialism in the nineteenth century, for instance, tied the idea of being white with high intelligence, high morality, and a higher sense of refinement. In other words, whiteness was associated with being socially superior to any other race of people. This led to widespread racism and discrimination.
Race, therefore, is so often tied up with cultural ideas and beliefs that sociologists have rightly advocated the abandonment of this term.