Haha. 李汉伦 pretty much explained it... In your situation, it sounds like the people are about to fight or compete."That's it! It's on!" can imply someone's about to fight the other, or that they're going to have a competition of some kind (sport, etc.) It can be used playfully, though. I'd recommend only saying it to friends. If I'm playing something with a friend to see who will win, I can say "Oh, you're on!" or "Oh, it's on!" but I'd have to show I'm joking by laughing or smiling or it could offend someone. Luck with your English.

lol, "it's on" is a relatively new phrase， similar to "bring it" or "bring it on" You use these when you are in competition with someone else or arguing etc. bring it (on) = bring your best (to the match) it's on = the "competition" has begun A and B are going to have a race tomorrow - A: You ready for tomorrow? B: Hell yeah! I've been practicing for months, if you wanna beat me, you best bring it! A: O, I'll bring it, don't you worry! C and D are debating something, it's about to get bitchy - C: You know it's true. D: Say what? that's a lie and you know it bitch. C: Why you gotta call me a bitch? D: And what? you are a bitch. C: O, that's it, it's on! D: Damn right it's on, bring it bitch! They fight. lol, PLEASE realise this is BAD English. Not only rude but shows very little 文化. Also, I've ONLY heard this from teenage American girls. This is not something that men want to be saying.