Ethnicity refers to classifications based on such elements as race, nationality, language, culture, religion, tribe, and background. Ethnicity also generally encompasses a large group of people who share these characteristics.
The most common identifiers of ethnicity are usually the most visible, so that would probably, at least in the United States, make race the most common ethnic identifier. The ability to distinguish ethnicity by color is more difficult now than it was, say, fifty years ago because of the dramatic increase in mixed-race marriages and the number of bi-racial children. A bi-racial President has, it seems, both heightened and diminished the significance of a person's racial heritage in the U.S.
In other cultures, tribe/culture is more significant than race, and there are countless examples of ethnic cleansing in which one group tries to eliminate (technically, to rid the country of) another group based on tribal, religious, or cultural differences.
Nationality is another kind of ethnicity. In America, this was a point of contention in many places with the early influx of immigrants and is still a matter of national debate regarding immigration (though the issue of illegality is probably more of a driving force than ethnicity).
Language is not a significant ethnic identifier in the United States, but Canadians, for example, have long experienced significant conflict regarding the two languages its citizens use (English and French).
Unfortunately, ethnicity is usually discussed in the negative, as points of division which create discord, confusion, or worse. The positive aspects of ethnicity include shared history, experiences, and cultural elements (music, art, clothing, food) by a group of people. The practice of classifying people according to ethnicity continues in polls and other information-gathering arenas; while this may serve to identify people in an attempt to reach them, it also serves to divide the country as well as various subsets, such as political parties and educational systems.