Civil rights are the rights that all people should have in a free democracy.
A civil right is an enforceable right or privilege, which if interfered with by another gives rise to an action for injury. Examples of civil rights are freedom of speech, press, and assembly; the right to vote; freedom from involuntary servitude; and the right to equality in public places. Discrimination occurs when the civil rights of an individual are denied or interfered with because of their membership in a particular group or class. Statutes have been enacted to prevent discrimination based on a person's race, sex, religion, age, previous condition of servitude, physical limitation, national origin, and in some instances sexual preference. 
Liberals and civil rights
In it's more traditional meaning, Civil Rights were distinct from Political Rights and Social Rights. Liberals were liberals because they advocated a liberal application of and liberal interpretation of those rights. Liberals believed that Civil Rights were a type of rights that were intrinsic in Mankind, the right to life, the right to Liberty, the right to own property, and the right to insure their safety.
Liberals still believe in universal Human rights, and those rights are in most ways the equivalent of the more traditional understanding of Civil Rights. Liberals still believe that those traditional civil rights are intrinsic in all mankind, and are appropriate for all people in all political systems.
Civil rights are what should prevent powerful people like Neocon governments or fat cat bankers or industrialists, Kings, Dictators or Theocrats from trampling on ordinary people. These are the types of rights that are written in:-
- The Constitution of the United States
- The Universal Declaration of Human Rights
- The European Convention on Human Rights
People’s civil rights are infringed if they are not treated equally because of their sex, their ethnicity, because they are gay, because they don’t follow the same religion as the dominant group in places like the United States Bible Belt. There also are more serious issues like freedom from slavery or involuntary servitude.  
Confusion and Controversy
The word Civil has come to have multiple meanings. Civility in social situations is not a civil right. Civil Rights are meant to limit the actions of Governments, Political entities, and those who enjoy special relationships with Governments or Political entities.
We may be Liberals, and we may prefer that all people treat each other with civility, but we absolutely do not claim government has or should have the power to compel good manners in a purely social situation. On the other hand, we believe that people shouldn't have the right to treat each other with incivility to the point of provocation, and have no right to insist that government protect their doing so.
People still have the right to think what they wish, to express those thoughts, or to be offensive. They just don't have the right to do so with the power of government.
As an example, the right to be legally married is a civil right. The marriage is sanctioned by government and conveys legal entitlements to the persons married. That particular power is given both to government and to Religious institutions. Government has the power to grant or recognize marriage. Government does not have the power to compel marriage, nor to compel Religious Institutions to grant or recognize marriages.
Now for the controversy part. Some governments have differing powers. In the US, governmental powers are limited. Other governments have the power to compel more aspects of public behavior than governments in the US do, and some people in the US would prefer that our government had more such power. We're Liberals, so we'd prefer a liberal interpretation of our rights, and that means a limited interpretation of governmental rights when it comes to our purely social actions. Some people would like to extend governments power into public places, like streets and parks and businesses.
Now, personally, I can see extending governmental powers somewhat in the case of streets and parks which belong to governments anyway, that's reasonable. Extending governmental power into business gets a bit more tricky. If the business is an artificial entity that can only exist by way of government, say a corporation or LLC, I might not object too vigorously to requiring civility of employees. However, both liberals and Social democrats agree to government regulation of businesses, for example compelling factories to provide a safe environment for workers.
Extending government power into a religious institution is specifically denied to government, and should stay that way unless the religious institution receives government support or tries to extend it's exemption into the marketplace. Extending government power to limit private people's freedom of speech, isn't possible in the United States but Canada and many European nations have laws against hate speech, Holocaust denial and the like.
In all cases of civility and civil rights, government has the power to grant some legal rights, but has only limited power to grant the expectation of civility. Even when Government has the power to grant the expectation of civility, we have to ask ourselves if it's worth the effort. Some changes need to progress slowly. If we try to mandate change too quickly, we risk creating such resentment that it will erode respect for government to such an extent that it's counterproductive.