What does the third wire on this speaker do?
The size difference and rough casing on the center wire indicate that it's probably shielded cable. This takes the form of a braided wrap on the two speaker wires, and prevents the speaker from receiving interference, for instance from 60Hz mains wiring or transformers in the iPod player device.

The speaker itself is basically an electromagnet, or, more simply, a coil of wire.  Current goes in one end, and out the other, and that's about it.  This coil should be isolated from the speaker casing.  The speaker is driven with AC, and probably capacitively isolated from ground.  Shorting to ground could damage the amplifier, depending on the configuration, so shielding is not generally used on speaker wire.  

However, in an alarm clock (plugged into mains), which is also used to play audio and charge iPod batteries (which are significant power draws), there's probably a sizeable transformer in the enclosure, which is probably pretty close to this speaker and the speaker wire.  This could produce a hum, so the designers of the device decided that the speaker needed to be shielded.  The risk of the speaker wire shorting out and damaging the amplifier is minimal, because the speaker wire never needs to flex within the alarm clock.

If you do connect these speakers to something else, you can probably ignore the third connection.  If you reuse the cable, simply leave the shield at the other end floating.