This means that a phone includes memory storage for MP3, AAC or similar music files, and software for playing that music. Generally, music can be downloaded into the phone from a computer and played back later through a headset attached to the phone.
This is not to be confused with real-music ringtones, which are usually a distinct feature. A "music player" feature can play whole songs, independently of the ringer functionality.
Newer phones with High-Speed Data may support downloading music directly over the mobile network.