A civil right is a right to freedom and equality that's guaranteed to every citizen. Your civil rights protect you against discrimination.
In the United States, individuals have the right to free speech, privacy, religious freedom, and many others, no matter their race, gender, religion, or sexual orientation — these are civil rights. When an employer refuses to hire someone because she's a woman or a landlord won't rent an apartment to a family because of their religious beliefs, it's a violation of their civil rights. The mid-20th century Civil Rights Movement was an organized fight to ensure the civil rights of African Americans.