migrate - Move from one country or region to another and settle there.
migrate is used for general movement.
Immigration and Emigration are specific type of migrations
emigrate - migrate from a place ( "E" in the Emigrate is like Exit, so it reminds me of exiting or leaving a place and that's how I used to differentiate between emigrate and immigrate)
ex. - Many people had to emigrate during the Nazi period.
immigrate - migrate to a new location ( "I" in the Immigrate is like In, so it reminds me of going into a new place)
ex. - Only few plants can immigrate to this island.
Hope this helps