It's totally understandable for the uninitiated to confuse the two since the drives look identical and so does the media.
CD ("Compact Disc") was created in the 1980's for music - these discs can hold about an hour of music or about 650MBytes of data.
DVD ("Digital Video Disc" or "Digital Versatile Disk" depending on who you talk to) was created in the 1990's for video. These disks can hold about 2 hours of video or about 4700MBytes (4.7GBytes) of data. They also come in "dual layer" versions that hold almost 9000MBytes (9GBytes) of data.
The oldest drives handle CDs but not DVDs.
Newer drives that handle DVDs are almost always backward compatible and work with CDs as well.
Drives that handle CDs will have the CD logo on them somewhere:
Drives that handle DVDs will have the DVD logo:
In addition to drives that read CDs and/or DVDs, there are drives that can "burn" (write data to) special blank CD and/or DVD discs. You generally need special software to do this (it usually comes with the drive) and there are a bunch of rules for how to do it depending on what kind of media you use and whether you want to play/view the disc on a standard consumer CD or DVD player.