Heaves is slang for COPD in horses. "Broken wind" is another term used, especially in the 19th century and earlier. I've also seen RAO (Recurrent Airway Obstruction).
COPD - Chronic Obstructive Pulmonary Disease - occurs in horses as well as humans. It's often caused by allergies and is generally seen in older horses - usually 12 and above. It causes a chronic cough and heavy breathing - sometimes horses have a "heave line" - extra muscle development along the flanks caused by the effort of breathing. Symptoms can range from a few coughs at the start of work to debilitating asthma-like problems.
Horses with heaves should be turned out as much as possible (in some cases they need to be moved from urban barns with limited grazing to more rural areas. Their hay has to be soaked and they often have to have special bedding to reduce dust. Some horses with heaves cannot be ridden in indoor arenas or have to be ridden right after the arena is watered. Medication is sometimes indicated.
Heaves is incurable and can be difficult to manage. In older eras a horse with "broken wind" would be put out to pasture or even slaughtered, and even now, some horses with heaves have to be retired from work.