The Nintendo Wii can't play movies from a hard drive without the use of additional software, but the newer Wii U can. People can use an SD card to record movies on a computer, and then play them on the Wii.
To play movies, download Wii Video from the shopping channel of the Wii. Then, use the menu to open the SD card, select a movie and play it.
To play movies from a hard drive with the Wii U, just connect the portable or external hard drive to one of the Wii U's USB ports. The Wii U can support hard drives with up to two terabytes of capacity. It also has an SD card reader like the Wii.