To stare is to fix your eyes on a subject. As hard as it might be to resist looking at them for long periods of time, it's not polite to stare at celebrities.
The noun stare is a certain kind of gaze — a long, intense one. The verb to stare is the act of staring, of locking one's eyes somewhere. When one stares at something, one looks at it for a long period of time, often with great intensity and without blinking. That's why one loses a staring contest when one blinks. Try not to stare at people when they're eating. It's rude! A stare is usually more intense and creepier than a gaze.