What could the red bump be in the corner of my eye?

See a doctor. Without examining you in person, it's tough to say what the red bump may be. It could be nothing or something more serious. If you have a suspicious lesion that won't heal, itches, bleeds, or is changing in any way, you should see your dermatologist (if it's on the skin by the eye) or an ophthamologist (if it's in the eye) so it can be properly evaluated.