“Come on, stay for one more round.”
You're out drinking with people. One of the people seems like he's ready to leave, but you think it would be more fun if he stayed for another drink. You suggest this.
Come on, stay for one more round.
When people drink together at a bar or party, they usually drink at about the same speed. So everyone gets a new drink at the same time. You call one group of drinks for each person a "round".
It's customary for one person to pay for an entire round of drinks for everyone, and then someone else pays for the next round. When it's your turn to pay for a round, you say:
This round's on me.
"Come on" is something to say when you're trying to persuade someone:
You also say "Come on!" when someone is upset or angry, and you're trying to get them to calm down:
Come on. Don't talk that way. Of course I love you.