Permission
The right to publish a copyrighted image is controlled by the copyright owner, so each copyrighted image that you use must have permission or fall within an exception, such as public domain, fair use, or open access. There are generally fees which come along with gaining permission to use a copyrighted image. These copyright permission fees are sometimes waived or reduced for scholarly publications; if not, however, they can be quite expensive as well as time-consuming to obtain. It is recommended that you begin the permissions process early to avoid any last-minute complications that may delay publication of your work. In addition to copyright permission, some museums and other providers of images charge a fee for the production or use of a digital image from their collections, even if the underlying work is in the public domain. Like permissions fees, use fees are sometimes waived or reduced for scholarly publications.
Keep in mind, the above recommendations refer to use of copyrighted images in a publication. You do not need to seek permission to use a copyrighted image for an assignment, as this generally falls under 'fair use'.
Fair use
According to the US Copyright Office: "Fair use is a legal doctrine that promotes freedom of expression by permitting the unlicensed use of copyright-protected works in certain circumstances." Essentially, fair use is an exception to the requirement to ask permission to use a copyrighted work, and its application is treated on a case-by-case basis.
Learn more about fair use.
Public domain
The term 'public domain' refers to creative work which are not protected by intellectual property laws, i.e. copyright, trademarks, or patent laws. As intellectual property laws differ by country and jurisdiction, creative works including images can be in the public domain in one country, yet protected by copyright in another country. In the United States, if you can find a usable image in a book or journal article published before 1927, the image is in the public domain.
Importantly: works of the United States government are all in the public domain, regardless of year of publication.
You can find more information on what materials are in the public domain in the United States by consulting this chart.