Apple's popular iTunes player will both create and play M4A files. iTunes will also rip CDs into M4A format. If your copy of iTunes isn't playing M4A files, you might need to upgrade to a more recent version. You can get a free iTunes player at http://www.apple.com/itunes/download/.
The Winamp player for Microsoft windows now plays M4A files. Support for M4A started with version 5.02. Visit winamp.com if you need to upgrade your version of the software.
The VLC Media Player is a free, open-source media player that plays M4A files. VLC Media Player is a good tool to have on hand as it is available for free for Mac, Windows and Linux and plays nearly every known media format, including .m4a, .mp3, .divx, .xvid, .rm, flac and many, many others. You can get VLC Media Player at videolan.org/vlc.
Many people have had trouble playing M4A files with Windows Media Player (WMP). There are plug-ins that should allow WMP to handle the files. Windows Media Player users should visit WMPlugins.com and search for .m4a to find the most recent plug-ins and codecs for WMP.
There are a wide variety of other programs that can be used to play, convert and create M4A files. The freedownloadcenter.com link in the resources section below will give you a long list of software that you can try for free. Another good source for audio software--one that is specifically mentioned by m4a.com--is AudioCoding.com, which contains free and open-source software for playing and creating a variety of audio file formats.