The white portion of the eye (sclera) can appear red when the vessels on the surface become enlarged. This may result from mechanical irritation, environmental irritants (such as extremely dry air, excess sun exposure), allergic reactions, infection, and other medical conditions.
A bright red, uniformly dense bloody area on the sclera results from a small amount of bleeding (hemorrhage) into the conjunctiva. It is a fairly common occurrence and is usually caused by straining or coughing. It generally clears up on its own after a few days.