If you aren't making your own original music and need music for your games or animation, you can't simply download your favorite song from the radio and use it. You need to find music from an artist who grants you permission to use it, or else find free-to-use music from a royalty free or no copyright source.
Here is a list of resources. Many still ask that you credit the source of the music in your work - which you should be doing anyway. When uploading music to NG, use the Credits feature to credit each individual artist, with links to their site. This makes it easier for our mod team to know your music is legit.
Newgrounds Audio Portal - Most musicians on NG will let you use their music if you ask them. There are also licensing details (such as a Creative Commons license) in the lower left column on each song page. Be sure the song isn't a remix or cover of someone else's song - if in doubt, ask the artist.
Incompetech - Kevin MacLeod's great archive of royalty free music.
Musopen.org - Royalty free music browsed by composer, performer, instrument, form and time period.
Free Music Archive - An archive of free music.
Audio Micro - Stock audio library with royalty free music and sound effects.
Epidemic Sound - Royalty free music.
W.O.W Sound - Royalty free music and sound packs with a focus on audio for games, including music organized by game genre.
Audionautix - Collection of music released under Creative Commons License 3.0 (free with no royalties).
Josh Woodward - Josh's collection of music he's made that you can use freely, just credit him.
Free Soundtrack Music - Collection of royalty free music, some free, some with a one-time fee.
Total Media Tracks - Royalty free music with reasonable one-time licensing prices and subscription access.
Premium Beat - Royalty free music that requires a one-time purchase fee.
TakeTones - Royalty free music with reasonable one-time licensing prices and subscription access.
YouTube Sound Library - YouTube's collection of royalty free music.
NoCopyRightSounds - A YouTube channel with copyright free music.