I am looking for an media conversion library that can convert and compress various media i.e both audio and video files to various formats.
FFMPEG-PHP is a popular choice for extracting information. It doesn't re-encode files, though. http://ffmpeg-php.sourceforge.net/
But if you have an instance of FFMPEG installed on the machine, you can call FFMPEG via the exec function in php. Eg: exec(’ffmpeg -i ‘.$SourcePath.’ ‘.$Destination);
I think your best bet is ffmpeg-php (can be used for both audio and video conversions). Imagemagick has a few basic video conversion options as well.
A second option would be to use mencoder from the MPlayer project (again, you'll have to call this as a command line tool). The main difference between ffmpeg and MPlayer is that the former comes with open source codecs while the latter comes with a host of codecs from all kinds of sources plus it uses ffmpeg.
So ffmpeg is a little more simple to use, MPlayer can convert between many more formats but the command line gets pretty complex.