I've had this issue several times before. Looking your dog in the eye always means that it's a slightly tense time between the two of you. Staring each other down is a means of determining domination. We all know this at some level. We see children trying this game out on each other regularly (I'm sure we've done it ourselves with other humans). We instinctively know that whoever looks away first lost somehow lost in some way.
If your dog looks at you in the eye and doesn't back down, it can often mean that your dog doesn't view you as the alpha person in the house. This isn't okay, since it can mean that your dog might not obey you and might act in violation of your wishes.
Ideally, a dog can look at you for a while and then drop his or her eyes after a while. Remember however, that you're not doing this to prove to yourself how powerful you are. You're just trying to reinforce the message to your dog that you retain the dominant post in your household pack.
If your dog stares at you, you must stare back. Chances are they will take the hint and back off or come and lick you.
Remember though to judge the mood. If you have something in your hand and your dog is just sitting happily, wagging her tail and looking at you, it doesn't mean that it's a game of dominance!