To emigrate is to leave one’s home country and move to another with the intention of living there permanently. It would be used in a sentence like this:
‘Having traced his ancestry, Mike found that his family had emigrated from Ireland in the 1840’s’.
To immigrate is to arrive in a new country with the intention of making it one’s permanent residence. Thus, if a newspaper runs a feature about high levels of immigration, it is referring to a high number of people entering the country.
To continue the example given above, ‘immigrate’ would be used in a sentence like this:
‘Mike’s family immigrated to Australia in the 1840’s’.
It all depends on whether you are speaking from the point of view of the country left or the country arrived in.
To migrate is simply to move from one country to another for an unspecified amount of time. Migration is the general shift and movement of people across the globe, without reference to whether they happen to be leaving or arriving.
The word ‘migration’ is also used to describe the movement of a particular species of bird or other animal from one climate to another. For example:
‘European swallows migrate to Africa during the winter’.
The key difference to remember is that emigration refers to leaving one’s country of origin whilst immigration refers to arriving in a new country. Migration is simply the act of movement from one country to another. Don’t forget that ‘emigrate’ is spelled with one ‘m’ whilst ‘immigrate’ has two!