If a blood vessel breaks under the surface of the eye this is called a subconjunctival hemorrhage and can make the eye very red, though it is a benign condition that gets better on its own.
Other possible causes of red eyes include:
1) Keratoconjunctivitis sicca, a dry eye condition that affects both the cornea and the conjunctiva.
2) Dysfunctional tear syndrome, when there is inadequate quality or quantity of tears (typically autoimmune).
3) Contact lense syndrome fro over-wearing or not properly caring for one's contact lenses, which can cause a build-up of irritating surface deposits or microbes (infection) of the eyes.
4) Computer vision syndrome from staring at a computer screen for too long, or one that is too bright.
5) Viral or bacterial conjunctivitis, an infection that may not cause pain or discharge in mild cases.
6) Uveitis, an inflammation of the middle layer of the eye, typically characterized by red eyes, light sensitivity and sometimes floaters.
7) Glaucoma, sometimes marked by a gradual increase in intraocular pressure (the internal pressure in your eyes) that can cause permanent vision loss wif not treated.
8) Other causes such as ciagrette smoke exposure, chemical exposures, chlorine exposures from swimming, or lack of sleep.
Because there are so many possible causes of red eyes, some serious, it is always best to have the eyes examined by an experienced healthcare provider and get on the proper treatment right away.