Nationality is usually easy – it refers to the state in which a person has the legal rights of a citizen. It is usually not controversial, but is occasionally where politics of statehood are difficult. A good example is Palestine and Israel; or a generation ago India and Pakistan. The international community tries to minimise statelesness, so everybody should have a nationality.
Ethnicity is a bit more controversial. Cultural self-definitions are often fraught and incompatible, and as a consequence, the way others see this is even more so. Two siblings may disagree on whether they are primarily Scottish, British, or Europeans, even if brought up in the same household and having similar experiences. Where ethnic identities cut across legal ones, situations can become very difficult – again consider Palestine and Israel, or the Indian subcontinent. A big part of the issue is that what criteria people use to understand one another differs from person to person – somebody who thinks language is more important would talk about a Punjabi ethnicity or a Bengali one, or the Anglosphere; somebody who thinks religion is more important may use ‘Pakistani’ as an insult to be hurled at a North Indian Muslim who in all other ways leads the same life as his Hindu neighbour.
Race is probably the most controversial of the three. Physical differences between people are obvious; however, the meaning of those physical differences is much less clear. Sometimes they are a reflection of genetic patterns, in which case the race concept is useful to, say, doctors; often however they are no more than skin-deep, so to speak. While in the past attempts were made to systematise race – for instance the old ‘four race’ theory that divided people into Caucasoid, Negroid, Mongoloid, or Australoid; or ‘craniometry’ which attempted to predict people's personality or intelligence characteristics by measuring the shape of their skulls – are generally classed as ‘scientific racism’ now, designed to justify and legitimise racist practice and colonialism. Further, as with ethnicity, what people think is relevant varies: some focus on skin tone, others on hair shape, still others on bloodlines.