Amnesty can mean a pardon for a wrongdoing, or it can also signal a government's willingness to overlook something.
Amnesty sounds a little like "amnesia," and that's because in its more specific sense amnesty means "forgetting." The government will essentially forget about whatever crime was committed, or whatever horrible things were said. As part of a truce, amnesty can be granted to opposition forces in civil disputes. Amnesty to illegal aliens means the government will deliberately overlook their illegal entry to the country. There can also be a period of amnesty when people can turn in something that they would otherwise get in trouble for. "The city offered a period of amnesty for everyone to turn in illegal guns."