Immigration is the process of moving to a new country to stay. New York's Ellis Island was considered a gateway for European immigration to the United States during the early 1900's. Nowadays, people immigrate through airports instead of islands.
The word migration is in immigration, which is something we think of birds doing when they fly South for the winter. It's not an accident — if American birds had little passports, they'd have to go through immigration when they got to Mexico. When humans immigrate, they have to follow the rules of the new country or else they risk getting in trouble. Immigration is the act of going to a new country, while emigration refers to the process of leaving one.