Irritants are just that - irritating. Especially for the sensitive mucosa that lines your nasal passages, sinuses, and on down into your lungs. One good example of a common irritant is tobacco smoke. The smoke from a cigarette has something like 4,000 chemicals in it. Many (formaldehyde, ammonia, and carbon monoxide for example) irritate the mucosal lining in your upper and lower respiratory passages.
In other words, these chemicals don't induce a true allergic reaction but they do IRRITATE the heck out of your nose and lungs. Of course many of these chemicals can cause cancer as well - so, smoking is a losing propostion all around.
Other irritants can include strong odors such as from bleach, or for really sensitive people perfumes can be an irritant. Pollution is another great example.