It is like an affidavit of sort, sworn out without the jurat and not before a notary. The swearing out of a complaint or rebutting evidence in all Federal civil matters (some states allow for the same) must contain an affidavit or an "unsworn declaration" that swears out the facts to be true and accurate, even though not notarized, and is based on fact and not supposition. It is subject to the same penalties of perjury if one lies as if you swear on a bible and testify in court or on a "sworn" affidavit. Affidavits need not be sworn before God, or on a bible. You have a right to just "affirm" that you will tell the truth, the whole truth, and nothing but the truth....and not "so help me God". Many courts don't use a bible at all anymore.
28 U.S.C. 1746 relates to these "Unsworn Declarations Under Penalty Of Perjury"
It is not b/c you don't believe in God that you'd use this...you always have the option of swearing out even a declared affidavit or testifying without swearing on a bible if you're an atheist. They just leave out the "before God" part.
Affidavits are the norm, however, in Federal Courts that have an expedited docket this is typically used when it could take a while to get a notary and the evidence is due. (In some states lawyers are automatically notaries but in others they aren't). The ability to swear out a complaint or contest a deposition without having to wait on a notary can be the difference between making your deadline or not. It's commonly used when records custodians are called to certify the authenticity of documents produced pursuant to subpoena or other formal request. Under F.R.Civ.P 56 declarations usually are not within the type of evidentiary categories that can be used at the summary judgment phase.
If it's a small misstatement you would probably be faced with a fine. If it's a total lie, outright, you'd be looking at jail time (say a records custodian removed evidence and swore out it was the complete business record in a fraud case). 18 U.S. Code § 1621 discusses perjury generally (in federal actions).