I am having back pain, slight shortness of breath, and I feel her moving very low in my pelvic and my entire stomach is hard.

We just had our little one, but things happened in such a way that we're still not sure what a contraction is.

The good news is you're "Full Term" which means that your little one can come at any moment. What you are feeling doesn't seem to be consistent with a contraction. It sounds more like pregnancy pain then anything. A contraction you will feel severe pain/tightness around your belly. If you can talk through it, it probably isn't a contraction. It will last for a little bit then roll away. They will occur with increasing regularity. IE, every five minutes, and the contraction will last a minute. Where you need to start heading to the hospital is when they're like that for over an hour.

You might be starting to go, but probably not. If you are then you're in such early stages that it will be some time before you need to go in.

"Entire stomach is hard" sounds like contractions, albeit mild ones if you can talk through them as CravenMorhead mentioned. I suggest calling your doctor to get their advice based on your medical history. For most, that might very well be early labor... or Braxton Hicks... but when in doubt, call the office ;)
Good luck!