Civil liberties are the sorts of things guaranteed to American citizens in the Constitution's Bill of Rights. I realize that's confusing because it's called the "Bill of Rights" rather than "liberties." But they are certain liberties or freedoms that are guaranteed to all: Freedom of speech, freedom of assembly, freedom of religion, etc, along with basic rights like the right to a fair trial, the right to vote, etc. Civil liberties are viewed from the standpoint of all persons equally.
Civil rights pertain to each person's right to operate as an equal in society and not be discriminated against on the basis of race, gender, etc -- in a wider array of situations than just the liberties assured in the Bill of Rights. For example, the Constitution does not give every person the right to be accepted into college. (That's not a guaranteed civil liberty under the Constitution.) But if a college is shown to discriminate against you and deny your enrollment because of your race or gender or disability or some other factor, then they have violated your civil rights to equal treatment.
