Most airlines are working on improving air quality inside their planes, but seasoned travelers know it is not the best. [Added since published: Newer planes like the Boeing 787 Dreamliner have advanced air management]
The air that you breathe inside an airline cabin isn’t nearly as good as what you will find most other places. The cabin is pressurized, since the air is so thin at altitude. However, they don’t pressurize it to sea level; it is substantially weaker. (This is part of why air travel can be so draining; you can end up with mild altitude sickness!)
Furthermore, the air is very very dry. This can lead to dehydration, which can also make you feel lousy. I strongly advise bringing water with you, and drinking water non-stop throughout the trip. If you consume alchohol, drink water along with your alcoholic beverage.
Finally, the air is filled with the exhaust products of your fellow travelers. One of these can be cigarette smoke; while smoking has been banned on domestic US flights, such rules are not followed around the globe. Although it might horrify some Americans, not all countries even mandate separate smoking sections!
Another exhaust product is germs. Because airplane tickets are so expensive, and because such effort is involved in making such a trip, people will fly sick. If your immune system is compromised in any way (HIV, chemotherapy, immunoglobulin deficiency), you may well want to wrap your face in a scarf or wear a surgical mask.