As someone else said, there are broadly 3 types of conjunctivitis - bacterial, viral and allergic. The common green/yellow matted discharge is bacterial. Chlorsig drops usually work for this. Viral conjunctivitis often accompanies colds (both can be due to adenovirus) and had a more clear, watery discharge. Allergic conjunctivitis is secondary to allergens and give a mucous, ropey, clear discharge. An all types the white of the eye (the conjunctiva) will become red.

Most conjunctivitis in children is viral, and as with colds there is no treatment. Artificial tear drops such as Refresh can help with the discomfort. If there is a green/yellow discharge, then it's probably bacterial conjunctivitis. You can let it run it's course, use breast milk, or Chlorsig drops. It can be difficult to diagnose for a lay person, so I would recommend seeing an optometrist can determine exactly what is going on. GP's vary in their eye knowledge, some are great though, but an optometrist deals with these conditions all the time.

A tip for getting drops in young eyes. Lie them down with their face up, let them close their eyes and place a drop in the inner corner of each eyelid. Then ask them to open their eyes (or you can gently open the lids yourself) while still looking up and the drops will get into the eyes.