The term "ethnicity," derived from the Greek ethnos ("nation"), is a fairly recent addition to the English language. The word only arrived on the scene in the 1950s (hence the lack of a Webster 1913 entry), when it came to mean a group of people sharing a common cultural heritage. This was a rather revolutionary idea, as it was always assumed in the past that people with a common cultural heritage also shared a common biological heritage - thus there was no need for the word "ethnicity" because there were already perfectly good terms such as "race," "people," and "tribe."
But unlike a "race" or a "tribe," words which imply genetic commonalities as well as cultural ones, a shared "ethnicity" is determined much more by the belief on the part of its members of a common heritage, rather than an actual shared line of descent. For example, the people today known as "the Japanese" are in all likelihood a mixture of several different peoples, so whereas calling them a single "race" is a questionable proposition, nobody would deny that they constitute a single "ethnicity."
For a precise definition of this oft-used term, I would be hard pressed to come up with a better one than the one put forth by Richard Schermerhorn in his book Comparative Ethnic Relations (1970). Ethnicity, he wrote, is "a collectivity within a larger society having real or putative common ancestry, memories of a shared historical past, and a cultural focus on one or more symbolic elements defined as the epitome of their peoplehood."