The Wii can play movies taken with most digital cameras without additional software by using the Photo Channel. If you have existing movies that were compressed in a format that the Wii does not understand, you can use Wii Transfer to convert those movies and automatically copy them to an SD card that the Wii can read.
To get started, you'll need an SD card writer. I recommend this SanDisk on Amazon if you don't have one. Once the movie has been converted and saved on the SD card, you insert the SD card into the Wii and use the Photo Channel to browse and play your movies.
After launching Wii Transfer, choose File → Convert For SD Card to open the SD card export dialog.
If the SD card has been inserted, you should see it's name appear in the popup menu at the bottom of the window. If you have multiple SD cards or other removable media (such as iPods), make sure the correct SD card is selected. Click the Choose button to find a movie to convert, then click the Save button and Wii Transfer will convert the movie and copy to the SD card.
That's it! The SD card is ready to be used on your Wii.
Demo limits: The demo version of Wii Transfer does not include sound in the converted movie. To remove this restriction, please purchase a full license for Wii Transfer. You will need to re-convert any movies after you register. See Purchasing.