It's very simple. Omegle is an IM client which puts you in random one-to-one chat with a person who is called stranger... another person just like you. At any point anybody can hit the disconnect button, the game is seriously minimal.
You can have some very weird conversations on here, and TBH I'm at a bit of an advantage considering my typing-fu is pretty strong, and I am prepared to use prepared munitions of copy-paste from monty python scripts.
It's a pretty good demonstration of the evils of the tower of babel principal. No name, no shame. Without a name, people talk quite differently.
I think this one was the most fun encounters I had in my little experiment with the site, the classic one-two takedown.
Connecting to server...
Looking for someone you can chat with. Hang on.
You're now chatting with a random stranger. Say hi!
Stranger: horny m 19 looking for girl to exchange nudes
You: I could just do a google image search and game you if you'd like
Your conversational partner has disconnected.