What muscles is important in controlling breathing?

The respiratory system is what lets you breath, but if you mean what controls your breathing, then your brain controls all of the automatic things we do like breath and blink.

Respiratory system muscles:

lungs

diaphragm

nose

pharynx

larynx

trachea

bronchi

The diaphragm is the most important muscle when controlling breathing.

The diaphragm contracts when you inhale to increase the space into which your lungs expand.