People stare at eachother as a way of communication. You might either show affection for the one you love or show the person in front of you you're listening to them. Dogs have had centuries growing closer to humans and their stare is often also a way of communication.
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They are waiting for us to communicate
When your dog is lying around or chewing a bone, they might follow you with their eyes. At these moments, they might just be waiting for you to knowingly or unknowingly communicate with them. They do not want to miss the moment you might pick up their leash or put on your shoes. These are signs for them they might accompany you on a walk. So by staring at you, they want to make sure not to miss any cue like this.
On the other hand, you might start asking them something to engage a training session. Because most dogs like the attention and rewards of a training session. Your dog might watch you hoping you engage in training with them.
They are trying to communicate to us
Ever noticed your dog sitting in front of the door with his head turned in your direction, just staring at you? They might want to tell you they need to go out. Or when he's just laying down next to his empty food or water bowl, staring at you, trying to tell you it's empty and they're hungry or thirsty.
Dogs that are taught to beg during dinner, also stare to communicate with you. They keep staring to tell you they want some of your food. However, this is taught behaviour. If you ignored your dog when he was still younger, they would have never benefitted from staring at you during meal times. Since you didn't, they learned that staring at you will result in them getting something of your plate.
Staring as a threat
When a dog stares at another dog without blinking with a very stiff and tense body posture and often accompanied by growling, this is a clear threat in dog language. Some dogs, most often if badly socialized or if they weren't cared for, can behave this way at humans too. If a dog is staring at you in this way, you'd be better to back off and leave the dog alone.
By getting to know your dog, it will become easier to figure out why they are staring at you. You will grow closer the more time you spend together and you'll notice you start to understand what he's expecting.
When staring at eachother, both in the dog brain and in the human brain, the hormone oxytocine is released. This is the love hormone which gives us a good and safe feeling. Because of this, staring can give both humans and dogs a good feeling, strengthening the bond between the two.