Abnormal breathing could indicate something like a heart defect, or an infection, respiratory distress syndrome (as from meconium aspiration), possibly low blood sugar...or other things. Basically, the point is if the baby is breathing normally, and has good color, then you can assume that baby is pretty healthy and has no major issues that need to be checked out immediately.
Baby should be breathing 'easy'--not making a lot of effort to breathe; no grunting with every breath tho some babies periodically grunt, groan, whimper, etc with a breath and some even hum with every breath, normally (I've recently seen a hummer...he is SO funny to watch!). No retractions should be present--when belly and/or area around throat and collarbones sucks visibly inward with each breath. Again, now and then the baby might retract somewhere with a breath--but this should not get more frequent nor happen with every breath.
Respiratory rate should be between 40-60 per minute--though it could be quite a bit higher for a few days and still be ok. Fast breathing is ok, if baby's color is good and baby can nurse, sleep, etc, without trouble. Naturally, babies breathe faster when having an exciting dream, or otherwise are excited--just like the rest of us...then breathing slows again with calming.
Baby might have blue/bluish hands/feet/nailbeds but should otherwise be rosy. Color should not get more blue when nursing (or anytime)--blue around the mouth when nursing (exerting) can mean heart troubles. Breathing should not get too difficult when nursing--tho of course, some babies choke on a strong spray of milk, and because nursing is work, they will breathe faster when nursing. But breathe-suck-swallow pattern should be coordinated without too much difficulty (if any).
I think that about covers it...
