After 9/11 Harold Finch - a reclusive hacker and billionaire - was hired to program a computer to find terrorists. "The Machine" spies on every person in the US and sends an alert when it picks up a terrorist plot. However, it can also pick up any other plot or bad intention.
Finch built a backdoor into the machine. Every week the machine outputs the social security number of a person who is either about to be killed or kill others.
Finch hired ex-CIA agent John Reese to protect these people and stop the upcoming murder. Reese, who is himself wanted by the government, co-opted two NYPD detectives to help him out: Joss Carter and Lionel Fusco.