Horses have pretty much the same respiratory organs and processes as humans. The biggest difference is size. This isn't surprising, of course, becasue such a large animal needs to be able to take in a lot of air.
When your horse breathes, it inhales air into it's nasal passages with the help of the diaphragm. The air passes through her trachae and into the lungs via the bronchial tubes. Here, the oxygen is taken from the air and transferred to the blood. The remaining carbon dioxide is exhaled back out through nostrils.
Horse's, just like humans, are susceptible to many of the same respiratory problems. These include allergies, chronic obstructive pulmonary disease and pneumonia.