Sometimes it gets a little tiring to work through all the details of copyright exemptions and exceptions. Especially when we're building on other people's stuff to make new things, we can make choices that make everything WAY EASIER - just search for inspiration amongst stuff you already know you can use!
(Note: this is all highly U.S-centric. Sorry, international folks, I don't know your public domain laws. Creative Commons should port, though!)
You Can Use Stuff When...Copyright doesn't apply
This is also known as the public domain - the body of works available for all of us to use. We all own them! Sometimes it's hard to figure out if something is in the public domain, but there are two times when it's fairly easy:
If it was published, in the United States, prior to 1923
If it was produced by the U.S. federal government.
There is a ton of other stuff in the public domain, including things from well after 1923, but that's the easy stuff.
ETA, 6/27/12: please note that neither one of these situations is always 100% easy! For the 1923 cut off rule, "publication" can be an issue (e.g., letters in an attic from 1912 - probably never published, possibly not public domain.) And many federal government resources are produced by contractors, who may own the rights. You may need to learn quite a bit more, to make determinations about public domain status for yourself.
If someone other than you is making the assessment of public domain status, you will want to consider the reliability and authoritativeness of that assessment.
Permission has been granted
Here's one way to know you have permission: "Can I use this?" "Yes."
This often works well when you can make contact with the creator directly, but note, this only works if you ask the person who owns the copyright - which is often not the creator!
Open licenses are another way to know you have permission - it's a form of licensing where the rightsholder gives the whole world a license to use the work (though sometimes with some restrictions.) Creative Commons licenses are the most common form of open content licenses; free and open-source software licenses are another example.
Places to Find Stuff You Can UseCreative Commons Search - http://search.creativecommons.org/
Search through many different sites for Creative Commons licensed content in a variety of media.
Internet Archive - http://www.archive.org/
Slightly overwhelming listing of multimedia content. Creative Commons (& other open licenses) and public domain.
"The Commons" on Flickr - http://www.flickr.com/commons
Photos from archives & museums with "no known copyright restrictions" (i.e., probably public domain)
Open Clipart - http://www.openclipart.org
Vector/clipart images. Widely varying quality. All public domain*.
Flickr Creative Commons content - http://www.flickr.com/search/advanced
Use the advanced search interface on Flickr to find only Creative Commons licensed photographs.
Musopen - http://www.musopen.com/
Collection of sheet music and recordings. All public domain*.
CC Mixter - http://ccmixter.org/
Collection of sound samples and finished musical works. All Creative Commons licensed.
Jamendo - http://www.jamendo.com/en/
Music site. All Creative Commons licensed. In and out of service, lately.
Magnatune - http://www.magnatune.com/
Music site. All Creative Commons licensed, but primarily available to fee-paying members.
Project Gutenberg - http://www.gutenberg.org
Collection of text and scanned-image copies of books. All public domain.
Open Library - http://openlibrary.org
Project of the Internet Archive - somewhat free e-books. Public domain and other.
Librivox - http://librivox.org/
Volunteer-read audiobooks. All public domain*.
Many authors have also released new books under Creative Commons licenses, but they're not easily searchable from any one location.
Al Jazeera's Creative Commons footage - http://cc.aljazeera.net/
Some of the better Creative-Commons licensed documentary footage available, though of course with limited topic coverage.
Vimeo - http://vimeo.com/creativecommons
Video site; allows creators to apply Creative Commons licenses to their videos. You can browse CC-licensed videos from this page.
Blip - http://blip.tv
Video site; there is CC-licensed content on here, but no easy search interface that I can find at the moment.
You can also search YouTube including "creative commons" as part of your search words - then check the full info to see if the video is licensed. (Or look at http://www.youtube.com/t/creative_commons for info on how to license, but they don't quite support the full suite of licenses.)
*For any new work, including present-day recordings of public domain music, copyright automatically exists from the time of creation. Although some creators or performers may want to relinquish all their rights, it's actually kind of legally difficult to do that - most countries, the U.S. included, automatically give some rights that creators cannot sell or give away. Several of the "public domain" resources on this list include new content, but creators on these projects usually at least intend to cede their rights.
Stuff to Help You Use This StuffThe following free and open-source software is all stuff I use (or have fairly recently used) on my personal computer (Linux Mint), my work PC, or both.
LibreOffice.org - http://www.libreoffice.org
Completely free and open suite of office software that interoperates pretty well with Microsoft.
VLC Media Player - http://www.videolan.org/vlc/
Audio/video player. Will play any format you throw at it.
GIMP (the Gnu Image Manipulation Program) - http://www.gimp.org/
Photo editor - like Photoshop, but free and legal to install on as many computers as you want! Very full-featured, slightly different interface, but if you give it a couple of days, the adjustment is easy even for a longtime Photoshop user.
Inkscape - http://www.inkscape.org
A vector image editor (like CorelDraw or Illustrator). Vector art is infinitely scaleable without loss of print quality, so this is a good tool for making posters, flyers, other one-page layouts. Most clipart (including all the .SVG files on openclipart.org) is vector art.
Audacity - http://audacity.sourceforge.net/
Great audio recorder, mixer, and editor.
VirtualDub - http://www.virtualdub.org/
A good stripped-down open source program for basic video editing.
Zotero - http://www.zotero.org
A very full-featured citation management program that is a browser plugin for Firefox, and that integrates with MSOffice and open office suites.
If you've got any favorite open content, public domain content, or open source software resources, feel free to add in the comments! Anon commenting is okay, but I'm a slow moderator. :)