Music player circuit
How much definition or detail will determine how much and what type of memory is needed.  Music could be played with methods varying from simple MIDI tunes requiring very little memory to varying compressions of soundwave files.  In some cases the same type of memory as you find in an SD card might be present, but not necessarily in SD card form.  Many types of memory exist that are suitable for low end audio.  For a simple "beep boop" type of song to play as little as a few dozen bytes might be used and varying types of rom would be possibilities.  

As for the type of circuit that can do such a thing, you need a device to produce vibrations, usually a speaker.  You need a device that can drive current through the speaker at different frequencies(essentially a processor), and you need something to store the list of frequencies to play at what time(essentially memory).  Many variations of each of these devices exist.

The tune may be recorded as simply as a list of notes and the frequencies to be played on that note and the duration.  Some of this may be done in hardware for a very simple device.  A wave file(uncompressed audio) is simply a list of amplitudes of the sound wave at increments of time.