An oath and an affirmation serve the same basic purpose: the person makes a serious promise that a statement is true or that a duty will be performed faithfully.
An oath may use religious language. An affirmation is a nonreligious alternative. The signer may choose either; the legal seriousness is the same.
When the promise is tied to a written statement and signature, the document may require a jurat. In other situations, the oath or affirmation may stand on its own.
I administer the words, observe the response, and complete a certificate when one is required. I do not decide whether the underlying statement is legally sufficient or factually correct.