Biologically, there are only three ethnicities: Asian, Black, and Caucasian. There are only three because there are three different shapes of the skull. For example, a person of Pakistani decent and another person of North Korean decent would both be considered Asian in this sense, in that they have the same skull shape. Same for someone from England and another from Brazil. People from Kuwait and Jamaica are also black in that they have the same shaped skull. Granted that the skulls of person A and B are not exactly the same, but they generally have the same characteristics, i.e. height, jaw joints, etc.
Socially, ethnicity refers to what country you hail from. Saying your American may be alright if you're in the United States, but in Spain or Australia they wouldn't know if you meant the US, Canada, Argentina, etc.
I don't think I need an example.