The Nintendo Wii is capable of reading DVDs because every game made for the Wii is pressed on DVDs but the Nintendo Wii cannot play DVD movies. Well a few hackers have been able to unlock DVD playback on the Wii. Symbiote has released a software player for the Wii, which requires users to have a mod chip installed in the machine.
The hack includes burning the application to a disc with CloneCD and then running it in the modded Wii. The Wii recognizes it as a Gamecube game and then allows you to remove the application disc to replace it with a DVD. Nintendo does plan to release a DVD player application for the Wii but for now you will have to go the illegal route if you plan on doing it (I would suggest just spending $30 on a cheap DVD player though).