No, the Wii is not designed or licensed to play DVDs. A complex process involving multiple steps and items and the installation of unofficial, third-party software is possible, but the DVD playback is not optimal.
The Wii does have channels available such as Netflix, Hulu Plus and YouTube, which are great ways to watch videos without using DVDs. Netflix and Hulu Plus require a paid subscription, but YouTube is available at no cost. If these channels are not already on the Wii home screen, they may be downloaded through the Wii Shop channel. If you are looking for a console that plays DVDs, consider the Xbox 360 or PlayStation3.