Dvdcd player is not reading some cds and reads others

This has to do with the type of CD you are trying. Is it a VCD, SVCD, etc.? Also, some CD players will only play certain formats of CDs. For instance, it will play regular store-bought CDs, and will play CD-R discs that you have made in the computer, but won't play any RW discs like CD-RW or CD+RW discs. ALSO, some players will play CD-R and CD-RW but any + discs, like CD+RW are not compatible.
Basically all players will work with a CD-R, and usually CD-RW but the + ones are less compatible. If I knew your model CD player I could check for you, but that is probably the case. Stick with CD-R and you should be good to go.
Lastly, rarely, but sometimes, the drive may not like the disc brand you are using due to the composition used (like if the bottom is blue or greenish). Tends to be that blue or what they say is cyan discs work best. I use TDK and Verbatim branded CD-R and CD-RW, and they seem to work best.
Good luck!!