One person making one move can be described with either word, depending on where the writer is standing. Someone who leaves Ireland for Canada emigrated from Ireland and immigrated to Canada. The prefixes carry the direction, and once you notice that, the pair stops being confusing.
A verb. To leave your own country to settle permanently elsewhere. Pairs with "from."
A verb. To come into a country to settle permanently. Pairs with "to."
Emigrate and immigrate describe the same move from two ends. You emigrate from the country you leave, and immigrate to the country you arrive in. Someone who moves from Vietnam to Texas emigrated from Vietnam and immigrated to the United States.
The first letters help. Emigrate starts with e, as in exit. Immigrate starts with im, as in in or into. The same split applies to the people: an emigrant is seen from the country they left, and an immigrant from the country they arrived in.
Which word to use depends on the point of view of the sentence. If it is about leaving, emigrate from. If it is about arriving, immigrate to. Migrate is the general word for moving, and it is also used for animals, seasonal work and, in business, moving data from one system to another.
His grandparents immigrated out of Ireland in 1950.
✓His grandparents emigrated from Ireland in 1950.
Leaving a country is emigrate from, with e for exit.
Thousands of people emigrated into the country last year.
✓Thousands of people immigrated to the country last year.
Arriving is immigrate to, with im for in.
The museum tells the story of emigrants arriving at Ellis Island.
✓The museum tells the story of immigrants arriving at Ellis Island.
Seen from the country of arrival, they are immigrants.
Quick way to remember
Look at the prefix. "E-" means out, as in exit — you emigrate out of a place. "Im-" means in, as in import — you immigrate into one. Whichever preposition the sentence uses, "from" or "to," will tell you which word belongs.
Can one person be both an emigrant and an immigrant?
Yes, always. It is the same person, described from the country they left or from the country they reached.
What about expatriate?
An expatriate, or expat, lives outside their home country, often for work, and usually without the sense of a permanent move that emigrate carries.