In theory, the NSA is not supposed to spy on Americans without a warrant. However, if they claim that they're really spying on a foreigner who just happens to be talking to an American, they can collect both sides of the conversation and put all the information in one of their giant databases. And once the American information is in their database, they can search for it because, technically, they're not collecting anything they aren't allowed to. They're just searching for stuff they've already "inadvertently" collected.
To you and me, this is laughable sophistry. To the NSA, it's just another day at the office. Today, however, the House voted to end these "backdoor" searches:
The House, by a 293-123 vote late Thursday, approved a bipartisan proposal to limit the NSA’s surveillance programs by requiring the agency to get a court-ordered warrant to search U.S. records in its possession.
....The FISA Amendments Act authorizes overseas surveillance of online and telephone communications and prohibits the agency from intentionally targeting U.S. residents. But the law does not prohibit the agency from querying U.S. communications inadvertently collected under the foreign surveillance program.
Intelligence officials have acknowledged in recent months they do conduct warrantless searches of U.S. records under the program, leading to protests from civil rights and privacy groups.
This hasn't made it into law yet, and I'm sure we can expect the usual process of watering down as the Senate gets involved and tricky little words get inserted here and there. But it's a start.