A writer / rewriter / burner is the same thing. When you create a CD or DVD you "write" it or "burn" it.
You often see the terns terms "R" and "RW" (such as CD-R and CD-RW).
"R" on it own means once you have written the data to the disk is cant be deleted. This is best when you want to make secure backups of your data that cant be accidently deleted.
"RW" on it means you can write to it, then later delete the data and write something else. This is useful if you want to put a film on it, then later delete it and put another film on it.
You will often see a "-" (minus) or a "+" (plus) after the DVD name (ie DVD+ RW or DVD-R) .
These were two different competing technologies that were developed when DVDs first came out (one company developed minus, the other developed plus).
Nowadays almost all DVD writers will support both minus and plus. Almost all DVD players will also support minus and plus. So you can use either really. I have always used minus, but that is more through habit than anything else.
However if you are going to create a DVD to "play" then make sure whatever you plan to play it on can support the DVD system you choose - minus or plus.