What is cat-back? and
What is full exaust?

The only thing I know is G35c have Y-pipe , mid pipe, and end exhaust. Are they called full exhaust or cat-back?

2 terms for the same thing. The axle-back term refers to anything after the mid-pipe.

Considering the location of our cats, there is no difference on our cars. Alot of cars have their cats towards the middle of the exhaust system...in which case a cat-back system would be the piping from the cats back to the muffler and a full exhaust would include the piping in front of the cats as well. Our cats are at the very front of the system though, so there is no piping in front of them, just the header.

a full exhaust is headers, test pipe (cats), ypipe, mid pipe, then muffler

a cat back exhaust is from the cats/test pipes back, y-pipe mid-pipe then muffler.