As a transitive verb, one which requires an object, oblige can mean to restrict by external force or circumstances. To be obliged is to be in someone’s debt because of a favor or service. As an intransitive verb, one which does not require an object, oblige means to take action as a favor, or without reward.
A person who obliges is an obliger, though the noun form is hardly ever used.
Obligate carries a slightly different meaning, which is to force someone (or an organization) to do something because the law or morality requires it.
Over the last hundred years, obliged has fallen in use while obligated has risen very slightly, though obliged is still more common.
One of the parties with a reporting duty is the housing company, which in this case gave the task to the property management company to take care of. In addition to the housing company, numerous subcontractors, ranging from companies removing asbestos to electric engineering firms, are obliged to submit reports to the Tax Administration. [Helsinki Times]
A member from their team went to different people at a restaurant and asked them for food, telling them how hungry he was. Not surprisingly, no one obliged him. [NDTV]
Those who have university degrees also felt less obligated to take care of their parents than those who never pursued post-secondary education. [Korea Herald]
Douglas County is not obligated to contribute the same amount to retiree health plans as it does for active employees, the Nebraska Supreme Court has ruled. [Daily Journal]