A dog staring at you or looking at you in your eyes is a natural dog instinct that usually means dominance.
The dog will usually stare into another dogs eyes for dominance.
When the dog stares into another dogs eyes the other dog has two courses of action which is either to gaze into the other dogs eyes, which may elevate the confrontation into a conflict, or avert their gaze, which signals submissiveness.
The dog could also be simply looking at you in the eyes because they like you or they are trying to get you to take them for a walk or play with them.
I had a dog that would stare at me into my eyes every time he wanted to go for a walk or needed to go potty.
So that was his way of telling me he wanted to go for a walk or go potty but it can mean different things for different dogs.