If you "owe (someone)" money, it means that you are supposed to pay that person:
He owes me twenty dollars.
Reasons that you might owe someone money include:
- you borrowed it from them
- you broke something that belonged to them
- you lost a bet
- you're buying something from them
You can also "owe" other things, like favors:
I owe him a favor because he helped me move a couple of months ago.
"Owing a favor" means that someone did something nice for you, so you need to do something nice to repay them.