The flash that David Hobby's using in that "cheap camera challenge" video is a Yongnuo YN-460--it has a built in "dumb" optical slave, which is a simple "see a flash burst--fire flash" sensor-based way of triggering a remote flash.
The 580EX does not have this capability. You can add the capability with a Sonia green-based optical slave, but then your main problem is that the 580EX also has no sync port to attach a slave to, so you'd have to get a hotshoe to sync port adapter or modify/replace the flash foot to include a sync port to plug the slave in. At which point you might as well just spend the same amount of money on cheap radio triggers (e.g., Yongnuo RF-603IIs).
What the 580EX has is a "smart" optical slave--one that speaks Canon's wireless flash signalling protocol, which is done with preflashes in the near-infrared range. To use the 580EX as a Canon wireless slave, you need a Canon wireless master unit. These are built into the pop-up flashes of the T3i (600D), 60D, 7D, and later cameras, but not in any of the earlier models. So, with, say, a T2i, you'd have to have a master unit on the camera hotshoe, and that would be an 90EX, ST-E3, 550EX, 580EX, 580EXII, or 600EX-RT. There are some third-party flashes from Metz, Nissin, and Yongnuo that can also be masters, but these are typically going to be the high-end expensive models.