The greatest compliment you can pay someone is to be interested in him or her and in what they have to say. Eye contact helps you to establish that relationship. Avoiding eye contact can give an impression of shiftiness, perhaps of lying. When you want to emphasise something you look the other person straight in the eye. Difficult negotiations are often described as 'eyeball to eyeball'.
Eyes have their own expressions.
Your eyes can fill with tears if you are deeply moved. They can narrow if you are wary, open wide if you are surprised. You can blink very fast if you are unhappy with the conversation you are having. Just as no one figure about a company will tell you how good that company is, so no one aspect of a person will tell you all about them. But the eyes are the second most telltale aspect of someone.