I had this same thing last year and I freaked out, thought it was serious and I went to the doctor and he said it was the blood vessel and it would be better in 2 weeks but it was all gone in 2 days after I went. I had it a total of 8 days so don't worry. On one side of my left eye it was red and I even had a tiny blood clot and it got worse, really red. I didn't do anything for it and it went away on its own. I got it from rubbing my eye a lot because of smoke getting into it. There are other ways to get it and I researched it to make sure I was not contagious and my doctor said I wasn't. I have it again now so I know it will be 7 or 8 days but my eyelid is slightly swollen and its a bummer all right but it will go away.
Hey chill out man, if it's just your eye that's red you probably just busted a vein in your eye, you're not gonna die from it tho so chill, I mean it'll go away after a while. But if you got stuff around your eye, thenn it's pink eye...and yeah that sucks /:
This is condition is usually due enlarged and irritated blood vessels in the eye and the first step you should take is to rest your eyes. Other actions to be taken include: not touching your eyes, not using make up and applying cool and warm compresses. You can also find more information from this link http://www.nlm.nih.gov/medlineplus/ency/article/003031.htm.