It's identity with a group of people who share a biological ancestor. All three display distinct cultural traditions that originate at particular hearths, but the three are regarded in different ways. Asian is recognized as a distinct race by the US Census Bureau, so Asian is a race and Asian-American as an ethnicity group are basically the same group of people. But, the Asian-American ethnicity lumps together people from many different countries in Asia. Most black Americans are descended from African immigrants and therefore also belong to an African-American ethnicity.
Some American blacks trace their heritage to Latin-American, Asia, or Pacific Islands.
The traits of race are those which can be transmitted genetically from parents to their children. For example, lactose intolerance is relatively common among persons of Asian, African, or Eastern European descent. It's extremely rare among people of Northern European descent. Nearly everybody is born with the ability to produce lactase, which enables infants to digest large amounts of lactose in milk. Lactase production slackens during childhood and almost disappears in adults, but a large percentage of persons of Northern European descent have a genetic mutation that allows them to produce lactase all their lives so they can continue to digest milk.