Discs that are 'R' are recordable only once
Discs that are 'RW' can be erased and reused
The '+' & '-' are different recording formats invented by different companies. Therefore some DVD recorders can only record one or other of the formats, although many PC drives can now record both. There is also a third format called DVD-RAM that is rerecordable, but very few players can even play these discs.
Once a disc has been recorded it can normally play on most DVD players, regardless of the origional format. There are however a few older players that wont work with some or even all of the recordable discs.
Discs are specifically '+' or '-' and need to be matched to what the DVD recorder is compatable with. If the recorder is compatable with both types then you can use either type of disc.
As for a couple of your earlier questions:
DVD-R(W) are very slightly more likely to be compatable with a DVD player, but the difference on modern players is very, very small.
You can keep adding video to a DVD disc until it is full, just like a VCR tape. Unlike a VCR tape though many players can only play a disc once it has been finalised and therefore nothing more can be written to it.