The major duty of the NSA that causes controversy is its duty to spy on communications of those outside the United States. NSA is responsible for capturing and analyzing foreign communications.
This can lead to controversy in at least two ways. First, it can arouse the anger of civil libertarians when NSA duties are carried out in the US. During George W. Bush's presidency, there was a great deal of controversy about NSA's warrantless wiretapping of communications between foreigners and Americans. Second, it can be criticized from the right if it does not do enough to help keep us secure. Conservatives will push for the NSA to be more aggressive in ensuring that (among other things) no further terrorist attacks can be mounted in the US.