You can very easily convert M4A music files with ease to more widely used MP3 format by using the same app that creates m4a files. That free application is iTunes which is the easiest way I know to convert m4a files to mp3 format. The best part is that it does not requires any additional downloads.
So, the first thing you will need to do is launch Apple iTunes app and then change the default type for imported audio. To do this, go to the iTunes Preferences and go to "General" settings. There will find an "Import settings" button. Click that and you will be taken to the screen that looks like the below given screenshot. Then, from the drop down menu, select MP3 Encoder and click the OK button. This will make the iTunes create MP3 files rather the M4A which is usually created. Now, you can easily convert all your existing M4A music collection with ease and that too directly inside the iTunes. Or if you wish to perform conversion on only a certain files, select the song (which is in M4A format) and right click on it and select "create MP3 version" from the drop down menu. [download itunes]