Someone who is obliged is bound to follow through on whatever is expected of him. Every April you are obliged or expected to submit your taxes. If you don't, the IRS is obliged to charge you with tax evasion. Ouch.
The word obliged comes from the Latin ligare meaning "to bind." Obliged typically refers to a moral or legal obligation or responsibility. If you're being followed by an ambulance with its siren on, you are obliged to pull to the side of the road and let it pass. In some cultures, women are obliged to cover their heads.