What gizmo do I need to play an MP3 through a car CD? Are they specific to each different kind of CD player or is there a generic gizmo?

You can get specific things that plug into the back of your head unit & enable you to connect 'properly' to your MP3 player.
The ones I looked at a while back were called something like 'Denison ICE Link'.

Or you can get an FM tuner that plugs into your MP3 player & puts out a radio signal at a certain frequency that you then tune your radio to. You get mixed results with these. I bought quite an expensive one that was supposed to have lots of features so the signal stayed clear & there were plenty of options for different frequencies etc. but it was rubbish. Unusably rubbish. I ended up taking it back.
But a friend has a cheap Griffin one that he keeps on one frequency & it is always fine – works really well.
I think to some extent it depends on the radio/aerial in your car as well as the FM tuner that you get.

Find out if your CD unit has an aux input line. If it does you can bring a lead out to the dashboard, glovebox or whatever and just plug your MP3 player into it

I've just had the same problem and ended up fitting a dension kit from ipodcarkitdirect. Worked for me to retain the current head unit. If you can change that then there may be other options. Dension kit works well for me though

There maybe a generic one but I recently fitted a Dension gateway 100 to my standard Golf stero.
It allows me to connect either an iPhone/iPod or any other MP3 player (via headphone socket). I use an old iPod and it's great – I can change tracks using either the iPod or the radio buttons and it charges at the same time.
Cost was about £130 and I fitted it in about 20 mins. I comes with a bit of wiring loom that fits into the back of the radio.

If you have an iPod then get a Harmon Kardon Drive+Play.

Work brilliantly and used to be around £130 but now you can get one for £30 odd!

Try Amazon

TBH, you might be as well looking at a new head unit. There are loads around with a direct input either from the front
or rear with a USB cable, and they are really cheap now too.

Try looking at some connects2 adaptors. The one I have lets me use the steering controls. They simulate the CD player but you need the right one for your head unit, oem stuff is usually catered for.