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:
Come on, stay for one more round.
You say "come on" to someone when you're trying to convince them in a friendly and casual situation. "Come on" means something like "Don't worry about it; just do it." Here is another example:
Come on! Tell me what she said!
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.
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