DVD-R is a record once format. It is compatible with all current DVD players and also with the vast majority of older players. Once the disc has been recorded the discs require finalisation to be able to play on other machines. Editing is limited and once finalised, no further recordings can be added to the disc, they cannot be re-used for recording and no editing can be done. -R discs use the VIDEO MODE for recording that mimics the structure of commercial DVDs as much as possible. This makes it the most compatible format for playing on other DVD players.
DVD+R is also a record once format, but it differs from -R enough to be incompatible with it. Once finalised it too can be played on all current DVD players, and again most older DVD players will also play +R. However some older players wont play +R. Like -R editing is limited, and they cannot be re-used for recording. +R discs use a recording format called DVD+VR, which is the same as that used on +RW discs but, as stated, editing is more limited. In theory finalising is not required but many DVD players will not play +R discs unless they are finalised.
DVD-RW is a re-writeable format that also allows fairly advanced editing. Pretty much all modern DVD players will play -RW discs once finalised, and some (noticeably from Sony, Pioneer and a few others) can also play them unfinalised. These discs can be re-used up to around 1000 times. To confuse things a little, -RW discs can use the Video Mode that -R uses as well as the more advanced -VR recording format. VIDEO MODE is more compatible with other players (especially older ones), but the trade off is editing capabilities. -VR mode allows far more advanced editing but is less compatible with other players, particularly older DVD players.
DVD+RW is the re-writeable format from the same people that invented +R. Features are similar to -RW and, once again, most modern DVD players will play these discs. In theory +RW discs do not require finalising, but in practice many player will not play them until they have been finalised. Editing features are close to those of -RW and they can also be re-used again and again. Like DVD+R, +RW discs use the +VR mode for recording but allow far more advanced editing.
Now, just to confuse matters even further, some multi-format recorders that can handle all the above disc types actually use the +VR mode even on -RW discs (Liteon recorders did this as did a few other makes).
For ultimate compatibility on other DVD players use DVD-R discs and finalise them once recorded. They are also extremely cheap but bear in mind you cannot re-use the discs for recording.
As others say, some recorders and players are fussy about the quality and/or brand of discs used (which essentially comes down to the dyes used in their production). You can sometimes find a brand that works well for years suddenly doesn't work in your recorder or player because the company producing the discs has changed supplier, or changed the dyes used in production. It can be a wee bit of a pain.