If you give someone power of attorney, that person, called the agent or attorney-in-fact, gains the ability to enter into agreements on your behalf. When your agent acts for you, he typically has to prove to others that he has your authorization to do so. Your agent can prove he is your agent by using an affidavit as to power of attorney.
A power of attorney lets your attorney-in-fact act on your behalf in any manner you allow him to do so. The attorney-in-fact typically interacts with third parties on your behalf. For example, if you grant financial power of attorney to someone to handle your business affairs while you are on vacation, your agent can pay your bills, conduct business and interact with anyone on your behalf as he takes care of your affairs.
An affidavit is a sworn statement made by a person who swears under oath that the statement is true. When you make an affidavit, you affirm that your statement is true and do so under penalty of perjury, just as if you made the statement in court under sworn testimony. Perjury is a crime. If you knowingly make a false statement in the affidavit, you commit the crime of perjury, and the state can punish you by charging with fines and incarceration.
When your agent acts on your behalf with another party, that party often requires the agent to prove he has power of attorney. A common way to do this is to require the agent to submit an affidavit as to power of attorney. In this document, the agent states that he is your agent, that you granted him power of attorney and that you have nor revoked it. The agent also typically includes a copy of the power of attorney along with the affidavit.
Not all third parties require an affidavit as to power of attorney, and those that do may make different requirements about what that affidavit has to contain. Regardless of the details, you must have the affidavit notarized or officially recognized by a person authorize by the state, typically a notary public. A notary is not there to confirm that what you state is true, but only to confirm that you are the person who made the statement.