Illegal immigrant — someone who has not entered a country legitimately, but has either done so clandestinely or lied to the authorities about his situation. Some economies rely on illegal immigrants for cheap labour. Illegal immigrants may include overstayers — people who remained after being refused leave by the authorities.

Refugee — someone who has been forced to leave his own country by persecution or impossibility of remaining there because of some natural calamity. Given the choice (including the removal of the source of persecution or sorting out the natural calamity) he’d rather go home. The reason for leaving his own country is more significant than the identity of his country of exile, i.e. the primary reason for going weighs more heavily than the choice of country of exile.

Asylum seeker — someone who found it necessary to flee his own country because of conditions there rendered life intolerable for him or members of his family. Such reasons may include persecution or targeting by kidnappers or other evil doers against which the police were powerless. They do not include economic pressure. Here the focus is on the country which has been chosen: he has left his own country with the intention of seeking right of abode in a particular destination. Under the various Conventions, he is supposed to seek refuge in the first ’safe’ country he comes to, but a surprising number arrive in the UK via all manner of other ’safe’ countries. For ’safe’ read ’soft’ (landing place).
References :

an illegal immigrant = Law Breaker

refugee and an asylum seeker =
someytimes a combination of the two above but not always.
References :

illegal immigrant – someone who does not have permission to be resident – ie may have come on holiday visa and stayed, or sneaked in

Refugee – some one who comes into country because they cannot live in their country anymore and have fled – religious persecution, war, racial prejudice, poverty and don’t want to return to own country

Asylum seeker – person who is officially recognised be governement of new country as a refugee and allowed to stay for set period of time (some not recognised and so stay as refugee but will be deported)
References :
http://en.wikipedia.org/wiki/Refugee
http://en.wikipedia.org/wiki/Immigrant
http://en.wikipedia.org/wiki/Right_of_asylum

Illegal immigrant – not gone through official channels in advance, and not from a country from which citizens may claim asylum.

Refugee – someone fleeing unrest or war in country of origin with a view to returning.

Asylum seeker – someone from a country or region experiencing political turmoil, who is asking for leave to stay here indefinitely on the grounds their life would be in danger otherwise.

Immigrants from other EU nations cannot be deemed illegal and cannot claim asylum.
References :
PS everyone – I can tell from the Q, this person is definitely British, so isn’t interested in the USA.