This is what we call a subconjunctival hemorrhage. While its appearance can be alarming, it isn't a serious problem. It happens when a blood vessel in the conjunctiva (the white part of the eye) breaks and releases blood. This often tends to occur after frequent sneezing, coughing, after rubbing the eye or for no apparent reason. It tends to happen more frequently in people who have high blood pressure or you use blood thinning medicines, such as asprin, motrin, plavix, vitamin e, coumadin, among others. It also tends to happen more when people have some redness in the eyes to begin with (blood vessels are already dilated). The blood will be reabsorbed into the body within a week or so. There is nothing you can do to make it go away sooner. Usually, there is a sensation of the eye feeling slightly full, or a mild pressure sensation and using artificial tears can help with this symptom. If you don't have any of the mentioned causes, and this happens more than several times, you should see your doctor to make sure there is no underlying ocular problem and to be sure you don't have high blood pressure or bleeding disorder.
I hope you find this helpful, Dr. Stevens
This link has a pic of a very small hemorrhage