Some burners can be very particular about what CDs they will work with, others will burn anything (mine do). If it starts burning for a few seconds and then spits it out, it's probably the CDs - try another brand.
If you have a burning program which allows you to get info on the cd (like Nero), check it and make sure the burner can "see" it. That often answers a lot of questions.
Also make sure your settings are correct and that you are actually asking it to burn a CD, not a DVD (easier to do than you think, he says :-[ )
There's also the possibility that (if you have more than one CD/DVD drive - especially if one isn't a burner, it's set to the wrong one (done that, oops) .
Try cleaning the CDs as well (they sometimes get a coating on them from being stored too long).
Have you been burning CDs before OK? This "will it won't it" situation can also mean a new burner is needed.
There are hundreds of possibilities depending on how much success you have had before.
Just because it burns DVDs doesn't mean the burner is fine, by the way nor does the fact that it plays CDs fine.
Finally, make sure you have the correct ASPI installed and that the various parts are compatible with each other. (Download ASPI checking program from Adeptec - Google for it).