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.