This is a lot more difficult question than it seems at first. I think of an ethnic group as people with a common heritage and culture, usually living as a minority within another culture. Jews in China or Argentina would be an ethnic group; Jews in Israel would not.
The term race is falling out of favor because it is impossible to define. It usually means a group of people who all have similar physical characteristics (hair color and texture, skin color, bone structure, etc. People once thought there were three main races of humans: caucasoid (white European), negroid (black Sub-Saharan African) and mongoloid (Chinese). All other types of humans were the result of the mixture of these so-called "pure" races. This is, of course, pure garbage, but some fools still believe it.
Nation usually means "country" like American, Mexican, British, Indonesian or French. Sometimes the nation doesn't exist like Basque or Palestinian. Which, by the way, are also ethnic groups.