An obligation is a duty. It's your obligation or responsibility to call your mom on Mother's Day. Sending flowers would be even better.
When you are morally or legally bound to a particular commitment, it's your obligation to follow through on it. If you see a crime taking place, for example, it's your obligation to notify the police. If an elderly person comes onto a full bus, it's your obligation to give up your seat for him. Sometimes when you enter a formal agreement with someone involving money, you might have to sign an obligation that states you are committed to paying back what you've borrowed.