Hello Molly, welcome to the forums.
The drive you are talking about is somewhere in the neighborhood of seven years old now. Originally it was designed to read and write both cds and dvds, but you have to understand that dvd drives have two lasers, one for dvds and one for cds. It is possible that this drive has suffered a partial hardware failure and cannot read dvds properly.
You can test it in another machine to confirm that.
You can also check a few things in this computer. This drive uses the older style IDE/PATA type of interface. There should be two separate IDE channels available to plug the drive into.
If the drive is on the same cable as the hard drive, try a different IDE cable and connect it to the secondary IDE channel.
IDE interface drives have to have a small jumper set correctly on the back of the drive. The options are Slave, Master and Cable Select. If you have the jumper set to Master, the drive has to be on the very end connector of the IDE cable. If you have it set to Slave, you need to connect to the middle connector. Cable Select allows the computer to find the drive and you can connect to either, but some computers are not very good at finding cable select positioned drives....that is rare, and is only found on very, very old computers however.