Broadly speaking, a civil right is a right created by a governmental body. For example, the right to free speech is guaranteed by the First Amendment of the United States Constitution. It is also protected as a right in many state’s Constitutions. Because this right is created by the Federal government, or your State’s government, it is a "civil right".
The rights that are most commonly referred to as "civil rights" involve the protections against discrimination on the basis of race, religion, gender, or national origin.
The most prominent example of a statute protecting civil rights is the Federal Civil Rights Act of 1964 which, among other things, prohibits discrimination on the basis of race in places of public accommodation. There are also many laws that make it illegal to discriminate in employment settings.