Race

The term race refers to the concept of dividing people into populations or groups on the basis of various sets of physical characteristics (which usually result from genetic ancestry). Your race is determined by how you look.

An example of race is brown, white, or black skin people  from various parts of  the world

Ethnicity

An ethnic group or ethnicity is a population group whose members are identified on the basis of common nationality or shared cultural traditions. Your 

ethnicity is determined based on the social and cultural groups you belong to

An example of ethnicity is German or Spanish ancestry or Han Chinese (regardless of race) 

Members of an ethnic group share certain language, heritage, religion  beliefs, values,and customs

You can have more than one ethnicity but you are said to have one race.

You can identify ethnically as Irish and Polish, but you 
have to be essentially either black or white

Race and ethnicity can obviously overlap, but they are distinct. For example, a Japanese-American would probably consider herself a member of the Japanese or East Asian race, but, if she doesn't engage in any of the practices or customs of her ancestors, she might not identify with the ethnicity, but might instead consider herself to be American.