To peer means to look at something with concentrated attention and possibly with some difficulty. Unable to find the light switch, she peered into the darkened room trying to find her coat. To gaze tends to imply that something is being looked at steadily or fixedly. Typically the thing being gazed at will be something pleasant. The travellers gazed in wonder at the vista of deep green valleys and high, snow-capped mountain peaks spread out before them. To stare has a similar meaning to to peer but it implies even greater concentration and fixed attention. If directed at a person this is the intentness of attention that is likely to be considered rude and to cause anger or apprehension. You may have heard of the "thousand yard stare". This describes the way a deeply traumatised person may stare blankly into the distance. The two young women, erstwhile chatting happily at the bar, fell silent when they noticed the way the man in the corner was staring fixedly at them. Hope this helps.

They are fairly synonymous, with slight changes in flavor. If I peer at something, I look intently at it with curiosity. If I look at something, don't know what it is and want to find out, I peer at it. 'Gaze' has a positive tone. If a woman gazed at me (I wish) it would mean that she was interested in me in a positive way. However, if a woman stares at me, her facial expression is somewhat negative and I might think that I'm wearing some piece of clothing wrong or spilled something disagreeable on my shirt.

"peer" means "to look into or up to". "I peered into the metal tube and saw the spider's eyes wink at me." "My peers always ask me good questions." ("peer" here is a noun meaning "person/people who look up to someone else to function as a role model") "stare" means to look intently at something. "I stared at the man's jacket and admired the cover art on it." "I stared at the woman's weird hairstyle." "gaze" is a stronger form of "stare" with a variety of meanings, but basically means "to stare with greater intent/to 'take it all in'" "I gazed as the sun set." "I gazed into the crystal ball." (call also replace "to peer")