Bouvier (1856) “An oath or affirmation reduced to writing, sworn or affirmed to before some officer who has authority to administer it. It differs from a deposition in this, that in the latter the opposite party has had an opportunity to cross-examine the witness, whereas an affidavit is always taken ex parte.”
Affidavits affirm information in front of an official witness and are often used to record statements that need some official weight but do not need to appear as evidence before a court where a deposition is more appropriate.
This example is simply attesting to the fact that J. D. Elliott, the affiant making the statement, knew the statements made in a particular petition were true. He makes his statement and signs his name. Like an oath, the statement is witnessed by a Notary Public with his official signature and seal.
In 1862, the U.S. government placed a tax on all legal documents (among other items including playing cards, financial transactions, and photograph prints) to help pay for the war. Notaries, court clerks, and judges sold stamps to citizens, affixed them to the documents as proof of payment, and wrote their initials and the date of payment on the top of the stamp to cancel it and ensure it was never reused illegally.