Quite simply, burst disks are one time use valves, extremely efficient at dumping air. Fast. It does what it's name suggests - once it reaches a high enough pressure, the flat disk material ruptures and releases the pressurized air behind it. For more specifics, visit the spudfiles wiki: http://www.spudfiles.com/spud_wiki/index.php?title=Burst_disk
The pressure at which a disk ruptures can vary depending on material and diameter. Aluminum foil is a decent, easy to obtain material and 1/2" disks will be used for this Instructable. Other things, such as plastic plates, duct tape, plastic sheets, and soda cans can also be used depending on the cannon design. A piece of tape 6mm in diameter can take over 300 PSI without bursting! You'll most likely want to experiment with materials and multiple layers if you create your own cannon design. To see a sample list of disk materials and breaking pressures, visit: http://www.spudfiles.com/spud_wiki/index.php?title=Burst_disk_material
Cascade failure is the use of taking multiple disks and letting them set each other off using a domino effect, taking advantage of pressure differences in a system. Because this is more complicated to explain in text, please view this link for more info and pictures: http://www.spudfiles.com/spud_wiki/index.php?title=Triggered_burst_disk
I recommend doing some research and skimming through the Instructable before building. It may be a good idea in case you wish to change the design.