The terms emigrant and immigrant are often incorrectly used, creating confusion at best, and annoyance of English teachers at worst. In general understanding the proper usage can help dispel confusion or quell the rage of would be wordsmiths. An emigrant leaves his or her land to live in another country. The person is emigrating to another country. An immigrant is a person who once resided somewhere else and now lives in your country.
For example, a Swedish woman decides to move to America. To herself, and to the country of Sweden, the woman is an emigrant to America. To her new American neighbors, the woman is an immigrant from Sweden, implying she has been somewhere else, and now is here, wherever here happens to be. Which term is used depends on whether or not she is being referred to from the US or Swedish perspective.
Emigration is the actual act of relocation from a country. The person going from one place to another is in the process of emigrating. The Swedish woman remains an emigrant to people of her country. To other Americans, she is an immigrant, because she has traveled from somewhere else.
During the French Revolution, people who had left France because of the escalating tension and violence in France were treated disparagingly if they returned to France. A person might be labeled an Emigrant, if he or she returned to France during the Reign of Terror or shortly thereafter. The term was meant to signify perhaps criminal behavior in fleeing France, as well as the fact that such people left the country.
When someone discuss his forebears who "immigrated" to the United States, he is in error. These people were emigrants to the United States. Since the person speaking is a US citizen, at least in this example, his forebears were immigrants, implying they had come from somewhere not "here."
In general, the distinction can be reduced to the prepositions "to" and "from." When a person moves to a place, he is an emigrant, but when he is an immigrant, he has moved from some place. Since technically an individual can be both, it makes matters quite confusing.
If the speaker or writer can remember that people "emigrate to" and "immigrate from," this helps to separate the two terms. It may also be helpful to realize that an immigrant is a new member of a society. An emigrant, on the other hand, is leaving his or her country.