Is my iPhone 5 that I bought online Factory Unlocked?

I recently purchased a factory unlocked iPhone 5 on the web, on the description it said "Factory Unlocked" and when it came in, and I tried to connect it to iTunes, it says "SIM Card Required" and I can't get passed that screen.

Does that mean that it is unlocked and that all it needs is a SIM Card?

I also have an old Cingular SIM, which I just discovered is not compatable with the current iPhones.

If I get a new Nano SIM, will ATT charge me for the data plan? & will I be able to stick to my current plan with my family?

Lot of questions so we need to break this down piece by piece.
SIM card required. Yeah, it'll need something to connect to your network. Without it, it's just an iPod.
Unlocked? No way of knowing until get past that screen. When you do go to Settings > General > About and look for Network and Carrier categories to see what is listed.
ATT? I think you are saying you already have ATT on a family package. They will typically give you one or offer for real cheap. Carriers don't really care what you use as long as the bill is paid. The problem is, if you are asking if ATT will charge you data. If you already have data on the family plan, then no. No changes will take place. If you don't have data on your current plan, ATT might be nosey about why you need a SIM card. Best luck.