Immigration describes a person entering a new country while emigration could be used to describe that same person leaving their old country. Immigration and emigration are both major factors that affects the size of a country's population.
Both words are based off of "migration," which means to move to one place from another. From the perspective of the United States, an immigrant would be anyone who enters the U.S. to live their life from any other country, like Canada, Mexico or Russia. From the same perspective, if a current U.S. citizen goes to live permanently in a different country, they have emigrated from the U.S.