Or - When in doubt, don't.
The only time that it is truly reasonable to post an unattributed photo or piece of artwork is when it's your own. Even if it's your own, consider including an attribution line, even if only to your username on Daily Kos, so that some future user has the option of giving you credit.
Photos that have aged out of copyright protection, photos and artwork that are in the public domain, and government photos from whitehouse.gov are all fair game for posting without attribution, but, hey, we still credit Shakespeare when we quote him, and a link to the site where you found the image is only courteous - and may save your bacon if you were wrong about the copyright status. (See below for a partial list of sources which provide "safe" images.)
Fair use for images is rougher than for writing, and writing is complex enough. If you're reproducing an image which is under copyright, which isn't licensed under Creative Commons, then your options are a) to reproduce it at a scale at which much of the fine definition of the original image is lost (somewhere around a 5:1 reduction from the original), or changed to the point where your changes constitute a substantial alteration from the original. And it's still good usage to either link or give attribution to the original source. Thumbnails linking to larger images, or stills from video, for the purpose of talking about the qualities of the larger work, are also legitimate usage. From Susan from 29, in the comments: The use of a book cover, with attribution, within a review of the book is considered fair use.
When you're in doubt about the copyright or republication status of an image, check the original site. Most sites that host artwork have a pretty clear statement of what secondary usage may be made of posted artwork. If it's still unclear, email the site, or the photographer, or the artist, and ask. If permission was needed, include a line which specifies it, and the attribution, and a link to the original site or image.
Hint: if you're not sure where an image was originally from, a quick way to check is to open a tab for Google Image, tab back to the image, and click and drag it over to the new tab and then to the "insert image" text. If there are multiple instances of the image, take the time to check the originals until you find one with attribution. Just because an image has been widely used without attribution doesn't mean it's not protected work.
Creative Commons (Note: this page has multiple links which clarify common usage of copyright terms, and it's worthwhile to check them out.)
Creative Commons Attribution 3.0 license http://creativecommons.org/...
You are free:
to Share: to copy, distribute and transmit the work
to Remix: to adapt the work
to make commercial use of the work
Under the following conditions:
Attribution: You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
With the understanding that:
Waiver: Any of the above conditions can be waived if you get permission from the copyright holder.
Public Domain: Where the work or any of its elements is in the public domain under applicable law, that status is in no way affected by the license.
Other Rights: In no way are any of the following rights affected by the license:
Your fair dealing or fair use rights, or other applicable copyright exceptions and limitations;
Apart from the remix rights granted under this license, the author's moral rights;
Rights other persons may have either in the work itself or in how the work is used, such as publicity or privacy rights.
Notice: For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to this web page.
AP, Reuters, and other major news organizations survive by selling and licensing their stories and photos. Reproducing this type of photo, even if you've seen it on a dozen different commercial or noncommercial sites, is a copyright violation, even if you source the photo. Even if a particular photo has gone viral, it's still an infringement.
Where to find Safe Images - partial list - feel free to add other sources you've found in comments, and I'll try to update as they come in.
Per Elfling: if you're a subscriber, it's generally safe to use images out of the DK image library. You can search using keywords. When staff licenses images for their front page stories, the license is good for the entire DK site, and included in those are paid-for stock photos and paid-for press images.
Government Photos
Government photos on the Whitehouse.gov site are not copyright protected. Third party photos on the site are available for use under Creative Commons Attribution 3.0 license. The same should be true on most government sites, but do check individually. A search for the term "image copyright" will generally find the information you need.
Wikipedia often has copyright free images, but always check and always attribute as they ask. If you're not sure, run a Google search to check. Wikipedia editors are NOT all knowing.
Public domain images from the Getty Museum: http://www.getty.edu/....
Press release photos - politicians usually have an official photo that is available and intended for use. Be wary of photos/artwork that are not of the candidate and/or not attributed directly - some political sites may use stock photos without attribution.
YouTube, Vimeo, Comedy Central, MSNBC videos, etc. - generally, if a site offers an option for offsite embedding, there are no direct copyright issues. If you're not sure whether the video might have copyright problems in itself, use a direct link to the hosting site, rather than doing an embed.
From Susan Gardner, in the comments: I also have been using this tool.
It's a Creative Commons search that lets you search several places at once ... Flickr, Wikimedia, Google images ... and supposedly only turns up works that are in the public domain or have a Creative Commons license.
A tip of the hat and a profound thank you to Elfling, who found time (I don't want to know how) to look this over.