The best free video software in the world. Unless specified, programs are available for both Windows and Mac.

Video Playback

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Miro
Beautiful interface. Plays virtually any type of video file. Subscribe to video RSS, download, and watch all in one. Torrent support. Search and download from YouTube and others.
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VLC
Plays more video files than most players: Quicktime, AVI, DIVX, OGG, and more. Pretty good interface.
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MPlayer
The interface and buttons are not as nice, but it may play even more video files than VLC.
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Perian (Mac Only)
A plugin that allows quicktime to play additional video formats. (Included in Democracy Player.)

Animation

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Handbrake
DVD ripper and MPEG-4 / H.264 encoding. Very simple to use.

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CDBurnerXP (Windows only)
Burn CDs and DVDs, including Blu-Ray and HD-DVDs, and burn and create ISOs. Despite the name, it does work on Vista.
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Burn (Mac only)
CD and DVD Burner. Burn any video to a disc that plays in your DVD player!

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Miro Video Converter
Convert any type of video to mp4 or theora. Convert any video for use with iPhone, iPod, Android, etc. Very clean, easy to use interface.
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Squared 5
Converter and encoder, with cutting, trimming, and joining.