Listening to everything

There are many agencies responsible for foreign intelligence collection in the US. The NSA does SIGINT, CIA is more operational and HUMINT, and there's also military intelligence and special purpose agencies like the National Geospatial Intelligence Agency.

I believe, officially, their role is foreign signals intelligence.

The NSA is under the defense department, they handle national security level information. As marco said, SIGINT and ELINT for example.

The NSA is our spy organization.

It's worth noting that the NSA is DoD while the CIA is State Dept.

They do have somewhat orthogonal missions (HUMINT vs SIGINT) as others have commented, but never underestimate each piece of the government wanting their own agents.