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.