Detonation cord is essentially a long thin bomb. Its purpose in life is to set off a group of explosives. For example, you often read about Navy SEALs rigging a number or explosives in a harbor and linking them together with det cord to set them off. Detonation cord is also a common prop on shows like Mythbusters because it is itself an explosive. According to this article:
The chief use of det cord, as stated, is to simultaneously detonate explosive charges across a fairly local area from a single detonator. However, its nature, being basically a tube of explosive compound, lends itself to the demolition of anything that the det cord can be wrapped around. For example, six tightly bound turns of det cord around a standard wooden telegraph or utility pole will neatly sever the pole. CAUTION, fifteen turns will not only cut, but will then launch the upper portion twenty feet in to the air! It should therefore be apparent that det cord is extremely handy for removing dangerous or diseased tree limbs, or the demolition of similar wooden members (wooden frame outbuildings or structures).
This video shows two meters of det cord at work on a tree:
You can see what det cord looks like and how it works in this video:
As discussed in the video, det cord is a thin, flexible plastic tube filled with PETN (aka Pentaerythritol tetranitrate). It is stable when handled normally, but sensitive enough to be set off with a bullet. With the PETN exploding at a rate of approximately 4 miles per second, any common length of det cord appears to explode instantaneously. You can see how fast detonating cord explodes in this video:
You can see detonation cord being used to set off a large number of explosives during a demolition project in this video: