Does it mean something is wrong with my eyes when only one of them water, swells, and gets irritated?

Conjunctivitis: This may be conjunctivitis, allergy or dry eye. Viral conjunctivitis migrates to the other eye within a few days, and is a self-limiting illness that will improve on its own in 1-2 weeks. Allergies often occur in both eyes unless the allergen falls into one eye only. Another possible cause is dry eye, which can be worse in one eye. If this is a frequent occurence, get checked out by an eye doctor.

Yes: If you have a red, irritated, watery eye, you need to be seen. There are many possible causes, some of which are serious. Don't take chances, see your eye doctor.