The FISA Puzzle
A useful starting point is the Foreign Intelligence Surveillance Act, the law they appear to be breaking. FISA sets up procedures for getting warrants to intercept communications between terrorists and Americans. The NSA is intercepting communications without such warrants. The obvious question is why--not why they intercept but why they don't get warrants to do it, either before or, as the act permits, after.
I can only think of two plausible explanations. One is that the wiretapping is on an enormous scale, perhaps of all phone conversations with one end in the U.S. and one end outside, using computers to convert speech into text, search the text for keywords and suspicious patterns, and forward the recorded conversation to a human being only on the rare occasions when that search reveals evidence that it may have something to do with terrorism. I think it would be technologically straightforward to apply that approach to, say, a million conversations a day--but I doubt the FISA court would be willing or able to issue search warrants on that scale.
The other possiblity is that the searches do not come close to reaching the probable cause requirement of both the Fourth Amendment and FISA. Checking every phone call might be a sensible tactic, but it would be hard to justify it under that standard. The same might be true of a much less ambitious program of interceptions, targetting conversations that there was some very weak reason to think might be of interest--say all phone conversations between the U.S. and any Muslim country, or all phone conversations involving a party who had been called by someone under suspicion, or by someone called in the past by someone under suspicion, or by … . The fact that someone has talked to someone who has talked to someone who has talked to an Al Quaeda member does not qualify as probable cause for tapping his phone--but it still might be worth doing, if the only cost is a few cents worth of computer time.
As I pointed out in a previous post, quoting a book manuscript that I webbed a few years ago, wire tapping by computer raises an interesting legal problem: If only a computer has listened to your phone call, have you been searched? One could argue that the search only occurs when a human being listens to the recorded conversation--at which point the key words found by the computer provide probable cause. If that is the Administration's unstated argument, it explains one part of the controversy--the issue of whether the NSA was using information from warrantless interceptions to get warrants for further interceptions.
I do not know if my conjectures are correct. What is clear is that any explanation of what the NSA is doing must explain why they chose not to follow the procedures laid down in the law. I do not see how any explanation that comes down to "wiretapping people we have good reason to think are terrorists" can satisfy that requirement.