Race is associated with biology, whereas ethnicity is associated with culture.
In biology, races are genetically distinct populations within the same species; they typically have relatively minor morphological and genetic differences. Though all humans belong to the same species (Homo sapiens), and even to the same sub-species (Homo sapiens sapiens), there are small genetic variations across the globe that engender diverse physical appearances, such as variations in skin color.
Although humans are sometimes divided into races, the morphological variation between races is not indicative of major differences in DNA. For example, recent genetic studies show skin color may drastically change in as few as 100 generations, spanning 2,500 years, as a result of environmental influences. Furthermore, the DNA of two humans chosen at random generally varies by less than 0.1 percent. This is less genetic variation than other types of hominids (such as chimpanzees and orangutans), leading some scientists to describe all humans as belong to the same race — the human race.
Ethnicity is the term for the culture of people in a given geographic region, including their language, heritage, religion and customs. To be a member of an ethnic group is to conform to some or all of those practices.
Race and ethnicity can obviously overlap, but they are distinct. For example, a Japanese-American would probably consider herself a member of the Japanese or East Asian race, but, if she doesn't engage in any of the practices or customs of her ancestors, she might not identify with the ethnicity, but might instead consider herself to be American.