For example, a Swedish woman decides to emigrate 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. So she has been an emigrant, in coming to America, and now she is a Swedish immigrant.
The term emigrant implies the process of travel. And emigration is the actual act of relocation from country. The person going from one place to another is in the process of emigrating. Our Swedish woman remains an emigrant to people of her country. To other Americans, she is an immigrant, because she has traveled from somewhere else.
Thus when we discuss our forebears who immigrated to the United States, we are in error. Our forebears were emigrants to the United States. To their country of origin, these people were emigrants. However, since we are US citizens, at least in this example, our 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 you are an emigrant, you emigrate to a place. When you are an immigrant, you have immigrated from some place. Since technically you can be both, it makes matters quite confusing.
If one can remember "emigrate to" and "immigrate from" this helps to separate the emigrant from the immigrant. As well it may be helpful to realize that an immigrant is a new member of one's society. An emigrant, on the other hand, is leaving one's society in search of greener pastures.