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!

Come on, stay for one more round.

You can also say "Come on!" when a person does something that you don't like, and you want them to stop. For example, if you and your partner were teasing each other, and he gets angry at you, you can say:

Come on! I was only joking.