Burst disks are a very simple mechanism for rapidly releasing a pressurized gas. They work exactly like the name implies; gas is held in a chamber with an outlet sealed off by the disk. The disk is in some way caused to rupture and all the gas is free to rush out the outlet (your barrel). They are appealing to spudders because of their low initial cost, fast opening, and excellent flow. In terms of performance, a burst disk is the best valve out there. They do have some major drawbacks though. It can be inconvenient to make and install a burst disk for every shot you want to fire, and it typically requires quite a bit of trial and error to make a consistent and reliable burst disk. Fittings that are typically used to hold a burst disk are unions, flanges, and cam locks. Commercial burst disks and holders are also availble.
Common methods of triggering burst disks:
Overpressure. The easiest and most common method of firing a burst disk gun is to simply increase the chamber pressure until the disk fails.
Mechanical. This simply involves mechanically shoving something into the disk hard enough to cause the disk to fail. A few common methods of puncturing the disk are pneumatic rams or dropping heavy projectiles with something pointy on the end down a barrel at a high angle.
Thermal. By heating the disk, the strength of the disk can be reduced to the point that it fails.
Pressure differential. This method involves using two burst disks with a small chamber between them; this chamber can be exhausted to fire the gun. For example you have two burst disks that burst at 70 psi and a chamber at 100 psi. If the small chamber between the two burst disks is at 50 psi then there is a 50 psi pressure difference across each disk and neither will burst. Once the small chamber is exhausted the rear disk will burst and allow the main chamber pressure to burst the front disk. A diagram can be found here.