There are some unique chemical compounds that burn by themselves - no oxygen required! This is because the chemical when energized, spontaneously decomposes and then the decomposition products "reduce," or change in ionic state by reacting with each other, as opposed to an outside oxidant.
The most commonly used monopropellant is hydrazine (N2H4), a chemical which is characterized as "strongly reducing". Another monopropellant is hydrogen peroxide, which when purified to 90% or higher is self combustible at high temperatures, or with a catalyst.
Engineers long ago realized the usefulness of monopropellant chemicals for satellite propulsion and attitude controls. Because only one chemical is used, the system is very simple, and thus very reliable.
Most monopropellant rocket systems consist of a fuel tank, usually a titanium or aluminum sphere, with a nitryl rubber bladder filled with the fuel. The sphere is then pressurized with helium, which pushes it out to the motors. A pipe leads from the bladder to a poppet valve, and then to the reaction chamber of the rocket motor. Usually, there's not just one motor, but two to twelve, each with its own little valve.
The attitude control rocket motors for satellites and space probes are often very small, an inch or so in diameter, and mounted in clusters that point in four directions. They look almost like toys.
The rocket is fired when the computer sends direct current through a small electromagnet that opens the poppet valve. The firing is often very brief, a few thousandths of a second, and usually sounds like a pebble thrown against a metal trash can. If the motor stays on for long, it makes a piercing hiss.
Unfortunately, monopropellants are not nearly as efficient as the other propulsion technologies. Engineers choose monopropellant systems when the propulsion needs are not very great. If the propulsion system must produce large amounts of thrust, or have a high specific impulse, as on the main motor of an interplanetary spacecraft, other technologies are used.