If you want to use an image for publication or for commercial purposes, you will need to determine whether the image is copyrighted or has other rights-use requirements such as licensing. If so, you will need to get permission before you use it.
Limits to Copyright Terms
Copyright protection does not continue indefinitely. Works for which copyright terms have expired are considered to be in the public domain, which means that you don't need permission to use them. A useful resource for determining if something is in the public domain is Cornell University's Guide to Copyright Term and Public Domain in the United States.
Outside of the U.S., copyright laws differ and you should not make assumptions based on U.S. law.
Copyright for Photographs of Artworks
With images, there is an added complication: an original work itself may be in the public domain, but a photograph of the work may not be.
For example, Michelangelo's sculpture of David (made in 1504) is well within public domain. However, a photograph of David made in 1975 may still be copyright protected; you would then need to get permission from the photographer or rights-holder to use it in a publication. In cases like this, it is always best to err on the side of caution and seek permission.
Copyright Permission vs. Use Permission
Depending on the image you've selected, you may need to get both copyright permission and use permission. The person who holds the rights to a work may not be the same person that possesses an image of the work.
For example, an artist might hold copyright to her or his artwork, while a gallery or museum owns the desired image of that work. You will need to secure permission from both parties in order to publish the image: copyright clearance from the artist and use permission from the gallery or museum.
This scenario is most likely to be an issue with living artists or artists who have been dead less than 75 years. In the latter case, you may need to seek permission from the artist’s estate. Again, this is the case for U.S. artists; copyright and artists’ rights differ elsewhere.
Increasingly, museums and cultural institutions make images of objects in their collection available free of charge for scholarly publication. Below are a few sources of cost-free images.
ARTstor's Images for Academic Publishing (IAP) program includes thousands of images available cost-free for use in scholarly publications; just add IAP to your search to show only images available through this program
Permission-Free Image Sources
From Wikimedia Commons. Photos by Eadweard Muybridge; animation by Waugsberg.
Here are some resources that feature copyright-free images for various types of use. Note that not all sites provide high-quality images, so they may not be suitable for some forms of publication.