I Can Breath Easily out of One Nostril but Can Hardly Breath at All out of the Other One?

Whichever side you can't breathe from has an obstruction that is permanent. You most likely have a deviated nasal septum that blocks most or all of your airway. Your nasal valve may be collapsed or the turbinate enlarged on that side. A simple exam with an otolaryngologist will give you a diagnosis and a septoplasty will most likely correct your breathing issue on the obstructed side. Once that obstruction is corrected, your normal nasal cycle should no longer bother you.

You are correct that the nasal cycle causes one side of the nose to be more congested and one side more open at any given time. The congested side and open side alternate approximately every four hours or so. Most people who have open nasal passages never notice the nasal cycle, however, because the breathing passages on each side are wide enough to tolerate the little bit of swelling that happens during the nasal cycle. When people start to be bothered by the normal nasal cycle, it is usually because their nasal passages are narrow at baseline, and when the small amount of swelling occurs during the nasal cycle, their nasal passage closes entirely on the more congested side. This narrowing may be due to inferior turbinate hypertrophy, nasal septal deviation, nasal polyps, or other causes. I recommend an in-person consultation with a board certified ENT who can do a proper intranasal exam and let you know what's going on. Good luck!