The vast majority of right-handed people are right-eye dominant, and the reverse is true for left-handed people. Of course, there are exceptions, and that assumption can't include everyone (though it does generalize most people into those categories). There are people out there who have opposite hand and eye dominance, which can be a little challenging when it comes to archery.
WHICH EYE IS YOUR DOMINANT EYE?
Don't know what your dominant eye is? Just want to make sure? Here's a quick and easy way to test your eye dominance:
If you hold your hand out at arm's length and make a circle, then view an object across the room by looking through that circle, your brain must choose which eye will actually focus on the object. Since your eyes are roughly 3" apart, both eyes cannot maintain the direct line-of-sight to the object. So one eye must take over, and you'll naturally position your hand more to the right if you're right-eye dominant, or slightly to the left if you're left-eye-dominant. In either case, the eye that takes over and maintains the sight-line is the dominant eye.
ONE OPEN EYE VS TWO OPEN EYES: So what's all this mumbo jumbo? What does it all have to do with shooting a bow? Sometimes it has nothing to do with shooting a bow whatsoever. If you plan or enjoy shooting your bow while closing one eye and looking through the other, it doesn't matter if your hand and eye dominance match or not. On the flip side, a majority of serious and professional archers shoot with both eyes open. Why? Shooting your bow with both eyes open allows for a larger frame of reference, and brightens things up - even when looking through a peep sight. Your dominant eye will focus onto whatever you're aiming at, while your other eye catches everything seen by your peripheral vision. In other words, it helps to fill in the gaps.