Snapshots are image files used to save certain moments of gameplay. They were introduced in Super Smash Bros. Melee; during a match, a player could use a virtual camera to take pictures. Taking a snapshot required one to enter Camera mode and use the controller in port 4 to control the camera. The snapshot would save to the GameCube memory card, leaving the feature rather limited; snapshots could be viewed in-game, but nothing else.
Brawl improves on the idea, and implements it in a much more useful fashion. Snapshots can be taken in any mode (except Stadium and Wi-Fi modes, although it's still possible to take snapshots from their Replays), using the pause screen's camera. Snapshots can be saved either directly on the Wii or to an SD card. The amount of pictures is only limited based on the size of the memory storage location, and pictures can be sent to friends via Wi-fi.
Note that the snapshot files when saved to the SD card are encrypted (so that Brawl doesn't read every image as a snapshot like the Photo Channel) and thus will need to be decrypted before they can be viewed on a computer. Decrypted snapshots wind up in .jpg format and appear squashed when taken in widescreen mode.