It sounds like you have a DVD-ROM drive which is just for reading DVDs (Movie DVDs, Music DVDs and DVDs with computer files).
To burn DVDs you need a DVD burner, the most common of which is called a DVD-RW Drive which are Rewriteable DVD burners.
A DVD-RW drive can burn DVD-Rs like what you have, but it can also burn DVD-RW disks which are rewriteable and so can be used over and over again unlike DVD-Rs which can be burnt only once.
Nearly all DVD Drives and DVD Players can use the DVD-R once you've burnt stuff onto it. To play it in a DVD player (i.e. through a TV) it would need to have audio or video on it.
I'm planning to get this drive [broken link removed] which has got good reviews.
One other thing to note is that virtually all DVD burners have CD burners incorporated into them as well.