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. You can read more about the Creative Commons here and here.
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.