To gaze at something generally means to look at in admiration or wonder, and is associated with looking at something for a long period of time because you enjoy it. It is never used negatively. However, to stare can mean to look at something for a long time both out of negative feelings and positive feelings, though it is generally negative. For example "I gazed at the stars in the night sky. It was a clear night, and they looked beautiful." or "The woman sitting across from me on the bus seemed to keep staring at me. It made me feel uncomfortable, and I kept wishing she would look away." Hope this helped!

To (hopefully) add to what Seth said - if you're looking at someone, a gaze would be relatively shorter than a glare (a second or two at the most), and less 'focused'.