There are two kinds of fuse: those used in explosives, and those used in electronic devices.
In explosives there are a variety of fuses, the simplest of which consists of a rope or cord which has been impregnated with combustible materials. When one end of the fuse is lit the flame will progress along its length at a predetermined and dependable rate. The other end of the fuse is, as anyone will remember from the Road Runner cartoons, attached to the explosive charge. The whole point of the fuse, and the importance of it burning at a constant, dependable rate, becomes obvious here. The person lighting the explosive will require time to achieve a safe distance (or in the aforementioned cartoon, the coyote will require time to ingest or somehow become stuck to the dynamite), and thus a sufficient amount of fuse is left to provide time for anyone in the area to reach safe distance.
The safety fuse, invented in 1831 by a leather merchant called William Bickford, was the first fuse to achieve this dependable timing. It was, compared to its predecessors, astoundingly consistent. Bickford, motivated to reduce the number of accidents in the mines, wrapped a core of black powder in textiles. Later on the cord featured an added waterproof coating, often asphalt, and later on a further outer sleeve of plastic or textile.
Black powder was eventually banned in many mines, and in 1908 the detonating cord made its debut. In many ways this was similar to the safety fuse, but contained high explosive rather than black powder. It was constructed in a rather unnerving manner, in that a large-bore cylinder of lead was filled with molten TNT and allowed to solidify before being passed, repeatedly, through rollers until it had reached the correct diameter. Called, in its native France, cordeau détonant, elsewhere it was merely called cordeau.
In 1936 the American manufacturers of cordeau developed their own version: Primacord. It was based on the French patents and consisted of a core of PETN or RDX surrounded by a variety of textiles, waterproofing and plastic.
In electronics, however, a fuse is almost the opposite; rather than causing explosions it is designed to prevent them, being a simple but important safety device. Imagine the bulb from a torch, in which a thin section of wire glows when electricity is passed through it. If too much electricity is passed through the bulb the section of thin wire will melt. The fuse works on a similar principle; a wire or strip of metal, designed to melt when excessive current passes through it, is placed in the circuit. When a potentially dangerous amount of current passes through, the metal in the fuse melts and the circuit is at that point disconnected from the mains. (This is usually a good thing, though occasionally one is led to wonder whether fuses are designed specifically to blow at critical moments during television programmes or when you're just about to finish the penultimate level of Tomb Raider.)