M4A is a version of the MP4 (MPEG4) media container format created for audio files. This container contains an audio stream that uses AAC (lossy compression) or ALAC (lossless compression) codec. M4A was developed by Apple and is currently used in its own iTunes store. There is also a variation of the format with the extension M4P, which includes copyright protection.
Most multimedia players support this format (Apple iTunes, Windows Media Player, etc.). Sometimes you may need to install additional codecs (for example, K-Lite Codec Pack). Since this format is highly specialized, sometimes to play it on devices it is easier to convert it into a more common format, such as WAV. This conversion will preserve the sound quality and allow to play the music on most devices.