An asylum seeker is someone who has fled persecution in their country of origin, has arrived in another country, made themselves known to the authorities and exercised their right to apply for asylum.
A refugee is someone whose asylum application has been successful and who is allowed to stay in another country.
A failed asylum seeker is someone whose asylum application has been turned down and is awaiting return to their country of origin. If it is not safe for refused asylum seekers to return, they may have to stay for the time being.
An economic migrant is someone who has moved to another country to work.
An illegal immigrant is someone who has arrived in another country and has no legal basis for being there.