Like other states mentioned here, AZ requires a background check (including fingerprints) and a one-day classroom course, with time on a range, to get a concealed carry permit. But, you don't need a permit to carry concealed in AZ--if you pass the federal background check, you can legally carry open or concealed.

My impression is that NYC and NJ (and perhaps other cities and states states) won't issue a concealed weapons permit unless you show an overarching need for one--i.e., you're in private security, or you carry large amounts of cash for business reasons. That requirement is being challenged in a federal court in California (San Diego, I think), on constitutional grounds--i.e., the sheriff or police chief should not have the right to set requirements for carrying.