General:
* Respiration
* Inhalation
* Exhalation
* Ventilation
* Aeration
More specific:
* Inhaling
* Exhaling
* Drawing breath
* Taking a breath
* Gasping (for labored breathing)
* Panting (for rapid, shallow breathing)
* Huffing (for labored, noisy breathing)
Figurative:
* Living
* Existing
* Sustaining life
The best synonym for "breathing" will depend on the specific context and meaning you want to convey.