In the U.S. any creative work - which includes images - is copyrighted from the moment it is "fixed in any tangible medium of expression, now known or later developed, from which they can be perceived, reproduced, or otherwise communicated, either directly or with the aid of a machine or device" [17 USC, Section 102]. Similar laws apply to websites hosted in other countries, so you should assume that every image - and every word, sound, and piece of software - you find on the Internet is under copyright.
However, those same laws allow copyright holders to give permission for various forms of reuse of their intellectual property. This Ars Techna post from 2011 is a good general guide to the use of Creative Commons images: Creative Commons images and you: a quick guide for image users.
The US Copyright Act lists six rights exclusive to the copyright holder:
It is not always required by the copyright holder, and not a legal requirement for using public domain materials at all, but it is good etiquette and a wise practice to always credit the creator of the image, music, or other intellectual property that you are reusing. In addition, you should always provide a complete citation for where you found the image or other information. If you did not create you should credit it and cite it, no exceptions.
The Creative Commons Search Page is an excellent first place to look for images because it provides access to sites that have images which are licensed for reuse. You can read more about what the Creative Commons does at their About page.
Enter a search term and select the kind of reuse you would like. Then select which site you would like to search. Your search parameters are 'auto-entered' into that site's search engine (sites need to be searched separately, and specialize in different sorts of images).
Each of the sites featured on the Creative Commons Search Page takes a different approach to displaying what sorts of reuses are permitted. If you cannot identify what sorts of reuse are permitted, please consult a librarian. As noted in the disclaimer at the foot of the page: "Do not assume that the results displayed in this search portal are under a CC license. You should always verify that the work is actually under a CC license by following the link."
Types of Creative Commons Licenses
If a work has entered the Public Domain, it means that the work "may be used freely, without obtaining permission from or compensating the copyright owner." Works can enter the Public Domain in a number of ways, including:
Check out the following resources for more information about the Public Domain: