In order to explain ethical obligations, one must understand each word individually.
Ethical is defined through the idea of what one considers to be right and wrong (or the concepts associated with what is considered to be right according to one's morals). Morals are the things which allow a person to make choices about right nd wrong based upon their individual mentality regarding a subject (like abortion, lying, murder).
Obligations are the things which a person is bound to do. Essentially, one is obligated to do certain things based upon either promises made or the convictions held.
Therefore, ethical obligations are the ideals a person holds considering what is right and what is wrong and the binds one feels/acknowledges according to the choices which should be made.
Essentially, a person who adheres to their moral obligations is examining their ethical ideas (what is right and what is wrong) and feels bound to do what is right (when concerning making a choice in life).
For example, if a person finds out that the company they work for are stealing from customers, the person's ethical obligation could be to quit or turn the company in. The fact that the person finds cheating ethically wrong will not allow them to continue to be associated with the company. Likewise, if a nurse sees a doctor acting inappropriately with a patient, the nurse's ethical obligation would be to report the doctor.