People owe things when they're in debt to someone. A friend can owe you a favor, and your employer owes you a paycheck.
When you talk about owing, it has to do with what someone deserves from someone else. If you borrow library books, you owe them to the library after a few weeks. When you take out a loan, you owe that money to the bank. You can also owe favors and more abstract things. If a friend did something bad to you, but they've been your friend a long time, you probably owe them a second chance.