I'm one of those people who still buy CDs. I use the digital files for mobile playback, but at home I like to use the physical ones. I purchased a CD-changer off eBay to store and play them without having to touch them any more than necessary. Sadly, it was broken when it arrived.
When it turned on, the blue light in the changer would flash and I could hear the motor running. It would make four attempts to turn the tray, running the motor for a few seconds and then waiting a few seconds. Then it displayed a "Table Error." I listened to it, but there were no grinding sounds, and it seemed like the motor was running fine. The disk tray just wasn't spinning. I could move it pretty easily with my hand (probably not a good idea, but I didn't break anything), but the motor wouldn't turn it. It worked when it was shipped, and it was pretty well-packaged, so I figured it probably got dropped during shipping and something got dislodged. Going into the project, I figured I could take a look inside and see if something was obviously out of place. I had very little to lose and if I didn't think I could easily fix it, I wouldn't worry about it.
UPDATE: According to jdros8822 in the comments, the replacement part number for the belt is SONY 421932601.
Step 1: Take the Cover Off
Unplug the disk changer, and take off the cover. There were 9 cover screws - 3 on the back, 2 on each side, and 2 on the top, near the front. Make sure not to lose these!
Once the screws were out, it's pretty easy to get the over off. I ended up sitting behind the changer and grabbing the edges at the back. I pulled outwards a little bit and then lifted upwards. With a little bit of wiggling, I was able to pull it backwards and off.
Step 2: Re-seating the Belt
With the cover off, if it is going to be easy to fix, you should be able to see exactly what's broken. For me, the band/belt had fallen off the pulley that was farthest from the edge (closest to the center of the disk changer). It's effectively just a thin rubber band on a plastic track, so it should stretch pretty easily.
Using a pair of needlenose pliers, (carefully) grab the band and lift it up. Once one side is on the track, you should be able to rotate the pulleys by hand to seat it all the way around, like you would do with the chain on a bicycle.
Step 3: Finishing Up
Wiggle the cover back on in the same way you got it off. It might take a couple attempts to get it seated correctly - I did it twice before I got it into generally the right place, and then had to wiggle the sides a bit to get them flush with the plastic front plate.
Put the screws back on and turn it on. Mine did a couple rotations, and then displayed "-NO DISC-" which demonstrated that it could check the entire thing. I put a CD in and played it to double check.