Textbook definition of race: a person's physical characteristics such as bone structure and skin color. Textbook defintion of ethnicity: cultural factors, including nationality, regional culture, ancestry-diffen.com

"Ethnicity vs. Race," diffen.com
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"The traditional definition of race and ethnicity is related to biological and sociological factors respectively. 
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Race refers to a person's physical characteristics, such as bone structure and skin, hair, or eye color.   Ethnicity, however, refers to cultural factors, including nationality, regional culture, ancestry, and language. An example of race is brown, white, or black skin (all from various parts of the world), while an example of ethnicity is German or Spanish ancestry (regardless of race).
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Definitions of Ethnicity and Race 

What is ethnicity?

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Ethnicity is state of belonging to a social group that has a common national or cultural tradition. This is, by definition, a fluid concept; ethnic groups can be broadly or narrowly construed. For example, they can be as broad as "Native American" or as narrow as "Cherokee". Another example is the Indian subcontinent — Indians may be considered one ethnic group but there are actually dozens of cultural traditions and subgroups like Gujarati, Punjabi, Bengali, and Tamil that are also bona fide ethnic groups. Yet another example is people in Great Britain — they may be considered British, or more precisely English, Scottish or Welsh.
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What is race?

A race is a group of people with a common physical feature or features. While there are hundreds — if not thousands — of ethnicities, the number of races is far fewer.
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Difference Between Race and Ethnicity
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Take the Caucasian (a.k.a., Caucasoid) race. The physical characteristics of Caucasians were described by M. A. MacConaill, an Irish anatomy professor, as including "light skin and eyes, narrow noses, and thin lips. Their hair is usually straight or wavy." Caucasians are said to have the lowest degree of projection in their alveolar bones that contain the teeth, a notable size prominence of the cranium and forehead region, and a projection of the midfacial region. A person whose appearance matches these characteristics is said to be a Caucasian.

Caucasians are found in many countries around the world. So while a Caucasian person in the United States may share certain racial characteristics with a Caucasian person from France, the two people have different ethnic backgrounds — one American, the other French. They will likely speak different languages most of the time, have different traditions, and may even have different beliefs that have been heavily influenced by their local cultures.
It is worth noting that "race" and "ethnicity" can be highly subjective, with lines between the two concepts frequently blurred. 

The video below discusses how terms for racial and ethnic identities have changed over the years and how a racial or ethnic term may not accurately describe a person's identity, as the person may have multiple racial and ethnic backgrounds."...

[You Tube, "Grey areas: The ambiguity of defining race and ethnicity," "Dr. Charles Hirschman, University of Washington Dept. of Sociology," Uploaded on Oct 27, 2010, "A visual sociology documentary on race and ethnicity at the University of Washington." Terms "ethnicity" and "race" are each a synonym for the other. He prefers to use the term "ethnicity," it's a form of identity, about where your ancestors came from. 2:19, "Race is usually thought to be a physical trait and ethnicity is thought to be a cultural trait. But they sort of slide into each other. A century ago the Irish were thought to be a race, the Jews were thought to be a race. At the present time, there are Middle Easterners, they're clearly a group that feels a great sense of identity and sometimes suppression. But they aren't necessarily recognizable by others."...]

(continuing): "Multiracial vs. Multicultural:" In most cases, race is unitary — i.e., a person belongs to one race — but may claim ethnic membership in multiple groups. For example, Barack Obama is racially black in spite of his mother being caucasian. On the other hand, a person can self-identify ethnically as Scottish and German if she has indeed lived in both ethnic groups.

Self-identification and Choice: Another difference between race and ethnicity is related to the ability to self-identify. A person does not choose her race; it is assigned by society based upon her physical features. However, ethnicity is self-identified. An individual can learn a language, social norms and customs, and assimilate into a culture to belong to an ethnic group....  
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"Asians"

The word "Asian" used in a racial context in the United States refers to people of Southeast Asian origin, including a vast variety of ethnic backgrounds, such as Chinese, Japanese, Korean and Vietnamese.

This colloquial usage is incorrect because "Asian" is not technically a race, as it means someone from Asia, including people from India, Saudi Arabia, Israel and parts of Russia.

Genetic Underpinning

Geneticist Luca Cavalli-Sforza has argued that neither ethnicity nor race have a genetic basis and cannot be scientifically defined. He notes that human populations have a great degree of genetic unity (even in their apparent diversity):
 
"Most genetic differences are between individuals, not groups. Almost never does one group (racial or ethnic) have a trait that is missing in the rest of humanity. Our physical differences—skin color, facial features, hair texture— actually represent ancestral adaptations to different environments. These are malleable characteristics that evolve relatively swiftly. The obvious differences in skin color, for instance, relate to the intensity of sunlight at different latitudes."" 
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