A music cd comes in a CDA format which gives you the best quality possible but the file is very large. When you copy it onto a computer the file is always converted into MP3 or WMA format, which is a compressed version of the music, so that it takes up less space. This allows you to put more music onto a standard music cd IF you do it in the compressed format. Most new cd payers can play MP3 and WMA files so you can go from home to car to office and still be able to enjoy your music, but older players do not have this capability and will not play the discs. This may, or may not, be your problem.
When you "burn" a music cd take care to pay attention to the dialogue that your software presents to you. Some programs ask you if you want to make a MP3 disc or the larger files of a CDA disc. When you choose the CDA format your computer will uncompress the MP3 files on your computer into CDA and then "burn" them onto a disc. They say that there is no loss of quality when you do this but I find that a little hard to believe. It is also possible that this step is where you may be having trouble with reading discs in your car. The files may be altered ever so slightly that your car stereo just doesn't recognize it. The only solution in this case is a new car stereo.
One other option, and one I have already chosen, is to purchase a MP3 player and a converter so you can play it on your tape deck or radio in your car. The Apple iPod is just such a device and it can hold upwards of 400 songs on the device. The price range is over $200 so it isn't cheap, but the payoff is never ending music arranged the way you want it, wherever you want it. I chose the RIO Nitrus MP3 player which goes for aroung $170 and has a capacity of 2.1 gigabytes. That means I can listen to music ALL DAY LONG and never hear the same song twice! The reason I chose this one is because it has a flash memory as opposed to the iPod's mini hard drive. You can bounce it, bump it or drop it and the music doen't skip a beat...no moving parts! The iPod will be subject to skipping because it is a spinning disc with a mini styulus that tracks the music. Later this year Apple is coming out with a flash memory version like my RIO Nitrus.
My sense tells me that in the near future cars will come equipped with some similar player function which will allow you to store music just like a MP3 player, but for now you will have to put up with cd cases all over the floor of your car.