Dogs stare at their owners for many reasons, but usually they are seeking attention or trying to understand their owner's mood. The latter is especially common in situations where dogs are nervous or uncomfortable and looking to their owners for cues.
It is usually a good thing when a dog stares at its owner. It shows that the dog is attentive to the owner's mood and cues. It is especially desirable during training sessions because the dog is ready to respond to the owner's commands at a moment's notice.
Dogs often use an intense stare as a way to try to attract their owners' attention. This is often done in the period leading up to meal times or when the dog is bored. This kind of stare can be harder to ignore. Some herding breeds use eye contact, known as a strong eye, to intimidate livestock and may try it with people, especially small children.
Although many people believe that dogs always consider eye contact to be a challenge or threat, it actually can be a bonding experience between owner and dog. Many dogs enjoy staring into their owners' eyes while being petted or held. However, this should only be done with dogs with which the human has established a good relationship. Unfamiliar dogs or dogs that are afraid of the person might consider it a threat.