rov1200
Good question.
From Wikipedia:
Leave to remain
The Leave to Remain is the legal status of a person issued by a government office of internal affairs to one who is not yet a citizen.
In most stable countries, Indefinite leave to remain (as is known in the UK) is granted to these foreign citizens after a specified period spent within its borders.
This is usually the final step towards naturalization which in turn leads to full citizenship.
A Leave to Remain need not be indefinite.
Amnesty
I couldn't find a definition of what this might be, but recently there was a debate started by Boris Johnson where he proposed an amnesty on illegal immigrants, who were able to support themselves and had been here for five years or more.
From www.londonist.com:
A sign of our twisted political times, perhaps, that our Tory mayor endorses an amnesty for illegal immigrants while the Labour government opposes it.
According to Boris Johnson, an amnesty for the estimated 725,000 illegals in the country would be "morally right", but the government, concerned about growing xenophobia and an impassioned (if often incompetent) effort by the BNP to sow discord, claims it would represent a "pull factor".
According to an LSE study, amnesty based on five years continuous residency could add 450,000 to the legal workforce, a group which Boris rightly points out we'd be better off taxing than spending millions of pounds to deport.
Yet with the economic crisis in full swing, and neither main party in support, it's a hard idea to sell.