Immigration is when someone comes to live in a country. eg. A German person comes to live in England.
Emigrate is when a person goes to live in another country. If you were to leave England to go and live in lets say Australia you would emigrate there.
Hope this helps
As LD suggests above, the process is just one thing and which word is used depends on which way you look at that process...rather like whether a glass is half full or half empty.
If a British person goes to live in Australia, we Brits think of him as an emigrant - an outgoer - and the Australians think of him as an immigrant - an incomer.