Is Iphone different from smart phone?  Which is cheaper?  I just need camera & calculator, nothing else.  Must I pay AT&T or ?? for those?

From the way you are typing i assume you are new to the smartphone market, I would recommend a cheap iphone as they are easy and clean to use. If you were a bit more tech savvy then i would recommend an android. I would perhaps recommend an iphone 5s/6.

An iPhone is a smartphone. All major smartphones do the same things, like calling, texting, etc. The main difference is in the software. Apple iPhones use iOS while most other smartphones run Android. The interface looks different, but the basic tasks are the same or very similar.

For your use case, buying an iPhone would be an absurd waste of money. You could even get by with a sub $100 Android phone. If you want to get the phone on a financing agreement, you will need to go through a carrier like AT&T. You will pay for the phone in monthly payments (unless you choose to pay outright) and for the plan.