"Making eye contact" with someone means looking at their eyes while they look back at you. When two people look at each other's eyes for a long time, you say that they're "staring at" each other. But when they look at each other for just a moment, it's called "making eye contact".
In the United States, it's polite for people to smile and make eye contact with people who they deal with. You should make eye contact with a customer, someone you've just met, someone you see on the street who you recognize from your work, and so on.