When a dog sits and stares at someone it's usually a sign of affection and love.
Most dogs sit and stares at people at people to express affection towards their owners and other people.
Your dog could also be wanting you to take them out for a walk or may be wanting you to pet them or play with them.
My dog will sit and stare at me and then glance over to his leash that is on the wall and lets me know that he wants to go out for a walk.
Or sometimes he'll bring me a ball and then sit and stare at me and glance towards the door and then stare at me again and give a little bark.
That means that he wants to go outside and play ball with me.