Assume all images are under copyright
Copyright is automatically granted at the time a new work is created, including works of literature, music, photography and images, and other creative works, and attaching a copyright notice to a work is not required. Copyright grants the author or creator the exclusive rights to display the work, in this case an image, to reproduce the work and create derivative works.
Fair use rules do apply, however, to images used in the classroom or for academic purposes. You can apply the four factors of fair use to analyze whether your use of an image might be allowed.
For specific information about proposed guidelines for using digital images for teach purposes, see Stanford University's Proposed Educational Guidelines on Fair Use.
If you are putting images on an open website, your safest option is to use public domain material, or use images with a statement of permissible use or open license (Creative Commons). Creative Commons licenses are a way for creators to specify the copyright restrictions for their works. These licenses are most often found on websites and images online. The licenses include Attribution, Share Alike, and/or Noncommercial, with or without allowing Derivatives. When searching the image website Flickr.com, you can limit your search to only photos with a Creative Commons license. You can also purchase images from websites such as Fotolia.com, Istockphoto.com, and 123rf.com. If these options are not feasible, you will likely need to obtain written permission to use image.