By default, no, the Wii cannot play DVDs straight out of the box. There are ways for it to play DVDs, however, but enabling the ability may void the warranty.
To enable DVD playback, you will need to install software to the Wii. The software uses an exploit, which is a security flaw, in the Wii's coding in order to make the Wii run unlicensed code. To do this may void your Wii's warranty, and you should only do so if you are confident that you know what you are doing.
The easiest way to install the software uses nothing more than the Wii and an SD card along with the HackMii installer. Bypassing these restrictions is not illegal in the U.S. for "homebrew" software, but using them to play copyright-protected video games is.
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.