DVD burner refers to a drive that can burn data/video to blank dvds. All modern burners can use +R or -R media, and they can use both single and double layer disks. Some can use DVD-Ram media, though this doesn’t seem to be a very popular choice due to the cost of the blank disks.
Not sure what dvr burner might be. DVR usually signifies a digital video recorder, so perhaps a replacement drive for such a machine?
DVDRom (when speaking about drives) signifies a computer drive that can only read dvds and cds, but cannot burn anything. DVDRom when applied to disks means a commercially made dvd that cannot be written to, only read.
Are you looking for a drive to fit into an external case that you already have? Or would you like a pre-made external dvd burner like this example: http://www.newegg.com/Product/Product.aspx?Item=N82E16827151194