The Android version of Minecraft does not include any of the game's music. Not only does this make playing the game less interesting, it also leads to bugs, such as the Jukebox seemingly not working.
The easiest way to get music working on Android is to simply access the Marketplace and download the official (free) Music pack. It should appear as the first result when you open the Marketplace and search for "music". We are also providing a downloadable zip file with a clean copy of the contents of that pack. These files are provided for convenience and compatibility only. Please buy Minecraft.
We have not yet tested a manual installation method for the vanilla music pack. By default, Minecraft automatically installs the pack to the resource_packs directory in it's app data. If your device is rooted you can easily just extract the zip file to that directory. If your device is not rooted, I suspect you'll have to figure out how to sideload third-party resource packs through your device's Internal Storage (/sdcard)