In every-day use, what do we mean by "racial" or "ethnic" identity, and why is it important?

To answer this question, let's first parse out the differences between ethnicity and race.
Ethnicity is a word we use to signify a person's belonging to a cultural group that shares something in common (such as language, religion, cultural traditions, place of birth, race, familial ancestry, history, etc). An ethnic group is based around similarities across people. It is possible for two people to identify the same way racially and a different way ethnically. For example, one person might describe their race as "black" and their ethnicity as "African American." Another person might also describe their "race" as black but identify their ethnicity as "Afro-Portuguese."
Race is a social construct, not a biological fact. When we label someone as a certain race, we are usually using their physical appearance to categorize them in a certain way. That physical appearance is often reduced solely to skin color. While there is no biological basis for these categorizations, society has ascribed meaning to them; consequently, racism has resulted as a way of justifying treating some people differently than others. Stereotypes have become attached to certain racial groups, proliferating a dangerous hierarchy of racial inequality. For example, America's legacy of slavery—in which people were bought, sold, and treated as if they were nothing more than livestock—persists in both cultural memory and social structures in modern America.
Race and ethnicity are important terms to understand because various labels in these categories come with certain privileges and certain stigmas. To be white in America is to be protected by virtue of your skin color. To be black in America is to experience a variety of stigmas, including ascription of intelligence, criminality, wealth, and/or education level, among others. To be Asian in America today is to be thought of as a smart, hard-working, and quietly-obedient citizen, but to be Asian in America in the 1800s was to be illiterate, unable to assimilate, and generally undesirable. While the racial stereotypes associated with being Asian have shifted over time, it's important to remember that any stereotypes (even seemingly positive ones) are an automatic reduction of a person's worth or ability.
Racial and ethnic identity can make us feel included or excluded, privileged or disadvantaged, free or oppressed. It can open or close doors to us in terms of education, class status, career, and social connections. It can bestow and advantage or a handicap simply by virtue of the fact that we were born into a certain ethnic or racial group. Race and ethnicity are the ways with which humans organize and categorize themselves, broadening or limiting a person's opportunities according to their identity. Additionally, the ways we identify ourselves and others are deeply impactful on our understanding of ourselves and others, as well as the way in which we experience and are experienced by the world.
https://learning.blogs.nytimes.com/2011/02/01/what-is-your-racial-and-ethnic-identity/