FFmpeg

FFmpeg is an open source software project whose main purpose is to convert video and audio formats. It also supports some still image formats. It includes:
 * libavformat, a library for reading and writing "container" formats and other file formats
 * libavcodec, a library of video and audio codecs
 * ffmpeg, a command line format conversion utility
 * ffplay, a simple media player

Libav
There is a fork of FFmpeg, confusingly named Libav. It is perhaps more focused on quality, while FFmpeg is more focused on features.