Rights refer to the privileges accorded to you by a governing body, and are usually written into laws; responsibilities are the obligations or duties that can either be assigned to you or assumed by you. Rights can be challenged or defended in a court of law.
The rights accorded to individuals or groups typically depend on an agreed set of obligations and behaviors, with the presumption of mutual cooperation and respect. Examples of legally defined rights include the rights of lesbians and homosexuals, the rights of minorities, the rights of children and the right to claim compensation in the event your rights are violated. Responsibilities make you ethically or morally accountable within the greater society.