An argument is circular if its conclusion is among its premises, if it assumes (either explicitly or not) what it is trying to prove. Such arguments are said to beg the question. A circular argument fails as a proof because it will only be judged to be sound by those who already accept its conclusion.
Anyone who rejects the argument’s conclusion should also reject at least one of its premises (the one that is the same as its conclusion), and so should reject the argument as a whole. Anyone who accepts all of the argument’s premises already accepts the argument’s conclusion, so can’t be said to have been persuaded by the argument. In neither case, then, will the argument be successful.
(1) The Bible affirms that it is inerrant.
(2) Whatever the Bible says is true.
(3) The Bible is inerrant.
This argument is circular because its conclusion—The Bible is inerrant—is the same as its second premise—Whatever the Bible says is true. Anyone who would reject the argument’s conclusion should also reject its second premise, and, along with it, the argument as a whole.
The above argument is a straightforward, real-world example of a circular argument. Other examples can be a little more subtle.
Typical examples of circular arguments include rights-claims: e.g., “I have a right to say what I want, therefore you shouldn’t try to silence me”; “Women have a right to choose whether to have an abortion or not, therefore abortion should be allowed”; “The unborn has a right to life, therefore abortion is immoral”.
Having a right to X is the same as other people having an obligation to allow you to have X, so each of these arguments begs the question, assuming exactly what it is trying to prove.