DVD players are usually marked DVD±R or DVD±RW compatible.
DVD-R (write): will automatically find free space on the DVD and record to it. Once done, the information can't be changed and the disc can only be played in DVD-R and DVD-RW compatible machines.
DVD-RW (rewrite): as above, but the information can be erased and recorded indefinitely, although the disc can only be played in DVD-R and DVD-RW compatible machines.
DVD+R (write): when you hit record the machine won't look for a blank space, but will record from the point the disc is at - like a VHS tape. Navigation through the disc's contents is via a menu with a thumbnail picture representing each recorded section. This format is compatible with most existing DVD players, but is not re-writable.
DVD+RW (rewrite): like a DVD+R, with the benefit of re-recording up to 1,000 times.
DVD-RAM: the information on these discs can be erased and re-recorded many more times than DVD-RW and DVD+RW. You can start watching a programme that's being recorded, pause live TV and watch one programme while recording another. Anything recorded will need to be played back on a DVD-RAM compatible player.
Blu-ray: a high-definition format that you'll need to view on a high definition TV to fully appreciate. Blu-ray discs have much greater storage capacity, so they can hold a film in much greater quality, although there are currently no consumer machines with a Blu-ray record facility.