There are several reasons an audio CD-R might not play, some of which are quite technical, and not at all obvious. I discuss a number of the reasons in my new article Why Can't I Play the CDs That I Burn?. It's the first in a planned series of digital music how-to articles.
An insidious problem is that the non-standard, high-capacity 80 minute, 700 MB CD-R discs won't play in many older players. What's worse is that they often get stuck, so you have to pry the player open to get your disc out.
The solution of course is to use standards-compliant 72 minute, 650 MB media, but it's getting very hard to find. I'm not able to find it anywhere - if you can tell me where 72 minute blanks can be purchased online, I'd be happy to link it from my article.