To say something is a “right” is to say that individuals are entitled to it, whether it be a thing or a form of treatment. People tend to use the term “right” as meaning that that the provision of that thing or treatment must be given priority, that by definition is deserved no matter what, and that there are no imaginable circumstances when something else should take priority over it. A condition of equal rights, then, means that each and every person gets the right that they, as individuals, deserve. And the individualist tradition central to American culture (to the point where many from all political points of view share in it) accords a central place to rights. The heart of a good society, many Americans (and many people around the world) believe, is the provision of equal rights.
That, however, is pretty much where agreement ends. What exactly does each person have a right to? To equal income? To a place to live? To quality health care? To free internet access? To marriage? Different people give very different answers to these questions. Also: how equally are individual rights now honored in our societies? Again there is strong disagreement.
A demand for “equal rights” in some form is currently a rallying cry for many who seek change in American society, whether they be liberals, progressives, or leftists. Often the focus of the arguments made is the claim that the poor, blacks, Latinas, LGBTQ individuals, and other members of minority groups are not treated fairly.
Many on the right who are sincerely committed to rectifying racial and gender injustices see today’s progressive claims for “equal rights” as having gone way overboard, to the point where they are a source of constant grievance that creates damaging hostility and resentment in society. From this point of view, current progressive and liberal demands are neither productive or fair, and in fact are draining and burdening society, not lifting it up and enlightening it in a positive direction.
Interestingly, though, conservatives invoke rights too (but different ones) to resist change, such as arguing for their right to practice their religion, to maintain the supremacy of traditional marriage, to protect their private property, to preserve their parental prerogatives. This has become stronger as they see cultural changes in a direction they oppose.