A solid-fuel rocket is a container with a flammable substance packed inside and an open nozzle at one end. When the substance is ignited, the exothermic reaction produces heat and copious amounts of exhaust gases. These gases escape through the nozzle so rapidly that a reaction force develops, and the rocket moves in the opposite direction from the exhaust gases (usually upwards).
The solid-fuel rocket has been around since at least 300 BC, when the Chinese histories first mention firecrackers. The most primitive solid-fuel rocket was a segment of bamboo filled with gunpowder (saltpeter, sulfur, and charcoal), sealed at both ends, and tossed in a fire. Whichever seal was weaker would fail first, and the resulting leak would be the nozzle. History mentions various improvements, but the first true rockets are mentioned in a text from the 12th century AD. These made a "sound like thunder" which could be heard as far away as five miles, and devastated large parcels of land. When the Mongols invaded Budapest in 1241 and Baghdad in 1258, they used similar rockets. The Arabs picked up the technology quickly, and used it against the French in the Seventh Crusade ten years later.
The state of the art didn't change much for the next 600 years. With firearms, a single soldier could carry almost a hundred shots with him, and fire with much more accuracy than the rockets of the day. After unimpressive performances in the War of 1812, the Civil War, and virtually every war in Europe up until the 20th century, solid-fuel rockets were briefly abandoned. Dr. Wehrner von Braun and Dr. Robert Goddard sparked a renaissance in rocketry with their liquid-fuel rocket theories, putting into practice what had been a pipe dream. By the end of WWII, liquid-fuel rockets were the undisputed favorites.
During the space race between America and the Soviet Union, the chemistry of solid fuels advanced significantly, and solid-fuel rockets were again useful, although they were still viewed as a little archaic. Their primary advantage over liquid-fuel rockets is that once they're built at the factory, they can be fired with little or no preparation--this is an ideal trait for strategic ICBMs. If a liquid-fuel rocket is fueled at the factory, the oxidizer typically attacks the metal tanks, and the lifetime of the fuel system components are immediately shortened, even if the fuel is removed again. Solid rockets, on the other hand, can sit on a shelf for years with no adverse effects, and modern propellants may last even longer--some of the newer ones have not been around long enough to gather reliable shelf-life data. One other benefit is the lack of hardware overhead: once a solid fuel rocket is built, it stays fueled until it launches. Liquid-fuel rockets, in contrast, are usually accompanied by all manner of (expensive) fueling trucks and gantries. One last advantage of solid-fuel rockets over liquid-fuel rockets is that for similar ranges, a solid-fuel rocket will be smaller and lighter.
Solid rockets typically contain a thick gummy polymer-based fuel which is cured in a large oven (yes, an oven, and yes, accidents happen, and yes, they're quite spectacular...) to the consistency of a rubber pencil eraser. While nobody ever mentions an oxidizer, there is typically a suspension of permanganate or another high-oxygen compound in the gummy base, so the "fuel" contains both the fuel and oxidizer. Aluminum dust is also included to increase the temperature of the reaction; this has the undesirable side effect of giving the smoke plume a massive radar cross section. Modern solid fuel rockets include the Space Shuttle's solid rocket boosters and the Atlas rocket's Centaur boosters. Weaponized solid fuel rockets include the Russian SS-16 "Sinner", SS-20 "Saber", SS-24 "Scalpel", SS-25 "Sickle/Topol-M", and SS-27 missiles, and the American "Minuteman" series, "Peacekeeper(MX)" series, and "Midgetman" series of missiles.