Well, there are really only two "families" of DVD at the moment (unless you want to count the red laser vs. blue laser stuff that's starting to come on to the market at the high end), and one "outcast". You've got the "minus" ("-") formats (DVD-R, DVD-RW) and the "plus" ("+") formats (DVD+R, DVD+RW), then there's the older DVD-RAM format that's harder to find these days. All of these are for data, of course -- all but the newest DVD video players will choke on those, in the same way early CD players choked on CD-R discs.
A DVD-R is a write-once format: once you've burned the data onto that DVD platter, the disk is forever frozen with that information. Add the "W" to that, and you'll find that DVD-RW can be erased or rewritten up to a thousand times. Seems kinda weird, but if you can do so, DVD-RW obviously has significant advantages over DVD-R. DVD-RAM was even more flexible, however, since it let you erase and rewrite sections of an existing DVD, something that you cannot do with DVD-RW.
Moving to the plus side is where things get a bit confusing, because DVD+RW came before DVD+R. The plus formats have the same data storage capacity as the minus formats (4.7GB), but DVD+RW offers faster writing, better internal linking (a technical obscurity you don't have to worry about), and support for drag-and-drop desktop files, which makes it easy to compose the contents of a disk. DVD+R is a write-once format intended to be more compatible with more DVD players, though at this point it seems to be about even with DVD-R, which remains the most compatible computer-burned DVD format.