Ans: 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.
In your case, since your drive is a DVD-RW, you're effectively limited to DVD-R and DVD-RW format discs. Stay away from any of the "plus" formats, as those won't work with your drive (and being newer, they cost more anyway). As to whether you want to use DVD-R or DVD-RW, that depends on what you want to use them for. DVD-R is a write-once format, just like CD-R, and you can't erase the data once it's written. DVD-RW is rewritable, so you can use it somewhat like a 4.7GB floppy disc, for all intents and purposes. In general, I suggest that you use DVD-R for archival purposes -- stuff that isn't going to change, and DVD-RW for more fluid data. DVD-RW discs are more expensive than DVD-R discs, so that may also influence your decision. How do you confirm what format your drive works with? One way, if you're on a Mac, is to use the System Profiler application. You'll find this useful utility in Applications -> Utilities. Launch it, then click on the "ATA" item on the list. If you have an internal DVD burner, you'll see something like what I get: "PIONEER DVD-RW DVD-106D".
If it's an external DVD drive, you might find it in SCSI, USB or FireWire, depending on how you hook it up. As far as your second question, your data-burning application definitely has to be aware of the DVD format you want to use (i.e. DVD-R or DVD-RW in your case). For Windows, you can use GEAR for burning CDs, but you'd need to upgrade to GEAR Professional Edition to be able to burn both CDs and DVDs. Toast Titanium does burn DVD-R and DVD-RW so you should be set in that department.