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.