As a general rule, if you didn't create something, you don't own it, and you can't photocopy it without the owner's permission. However, as with every rule, there are some exceptions.
You can copy your own official identity documents as long as there's no intent to commit forgery, and you can copy a small excerpt from copyrighted documents for "fair use" purposes, such as teaching the material in a classroom setting.
TL;DR (Too Long; Didn't Read)
It's illegal to photocopy U.S. Mint paper money, Treasury checks, unused postage stamps, copyrighted material and another person's personal documents, with limited exceptions.
Currency, Stamps and Treasury Checks
It's illegal to photocopy full color, full-sized, double-sided copies of United States paper currency and U.S. Treasury checks. You could be committing the crime of forgery if you do so with penalties ranging from probation and community service to five years in prison and a $125,000 fine.
It's acceptable to make a one-sided copy of paper bills that's less than 75 percent or more than 150 percent of the original size, however. If you're copying unused postage stamps, make sure they're black-and-white copies or color copies that are no more than three-fourths of the original size.
It's not illegal to copy your own identification documents except for purposes of fraud. In fact, the Department of State recommends that you take copies of your travel documents in case of emergency.
It is illegal to photocopy someone else's records, however, unless that person has consented. Businesses routinely photocopy passports, driver's licenses, birth certificates and bank statements as part of their background checks, and this is fine as long as they have your consent.
It's illegal to copy books, magazines, documents, photographs, artwork and other copyrighted material unless you have written permission from the author or copyright agency. But you can copy small excerpts under the "fair use" laws for the purpose of research, teaching, journalism or to criticize, comment or parody the work.
Fair use is a defense against a claim of copyright infringement. You must show that you've copied only what you needed to make your point, and you must give credit to the original source.
Public Domain Documents
Copyrights don't last forever. After a specific period – usually the life of the author plus 70 years – a previously copyrighted document will fall into the public domain. Some creators even dedicate their work to the public domain before the copyright period expires using a Creative Commons license. Although it's perfectly legal to copy a public domain document, you should always assume that a work is copyrighted unless you can prove otherwise. The U.S. Copyright Office catalog is your first stop for researching copyrights online.