There are many different types of ways to accelerate craft in space. Below is a summary of some of the more popular, proven technologies.
Solid
The first rockets made by the Chinese used gunpowder, and until this century all rockets used some form of solid or powdered propellant. Solid rockets are considered to be safe and reliable; these kinds of rockets have been used for so long that engineers have a very good understanding of them.
One of the many drawbacks to solid rocket systems is that once they've been ignited, they can't be turned off. That is why they aren't commonly used in satellites; satellites need rockets that can be fired multiple times (or pulsed.)
Hybrid
A hybrid propulsion system is composed of solid fuel and liquid or gas oxidizer, typically. These systems are superior to solid propulsion systems because of safety, throttling, restartability, stability, and environmental cleanliness. Hybrid systems are more complex than solids, though, and more expensive.
Monoprop
Most familiar combustible chemicals (gasoline, alcohol, wood, natural gas) require oxygen to ignite and burn, but there are some unique chemical compounds that burn by themselves - no oxygen required! This is because the combined chemicals, when energized, "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".
Rocket scientists long ago realized the usefulness of monopropellant chemicals for satellite propulsion. Because only one chemical is used, the system is very simple, and thus very cheap. Unfortunately, monopropellants are not nearly as efficient as the other propulsion technologies. Engineers choose monopropellant systems when the satellite's propulsion needs are not very great. If the propulsion system is going to be heavily used, such as on a geosynchronous satellite or an interplanetary spacecraft, other technologies are used.
Biprop
Systems that use both a propellant and an oxidizer are called bipropellant systems. Bipropellant systems are more efficient than monopropellant systems, but they tend to be more complicated because of the extra hardware components needed to make sure the right amount of propellant gets mixed with the right amount of oxidizer (this is known as the mixture ratio.)
Dual mode propulsion
Dual mode propulsion systems combine the high efficiency of biprop with the reliability and simplicity of monoprop systems. Dual mode systems are either hydrazine/N2O4, or MMH/peroxide (the former is much more common). Typically, this system works as follows: During the initial high-impulse orbit-raising manuevers, the system operates in a bipropellant fashion, providing high thrust at high efficiency; when it arrives on orbit, it closes off either the fuel or oxidier, and conducts the remainder of its mission in a simple, predictable monopropellant fashion.
Resistojet
A resistojet can be thought of as a "space heater attached to a water hose". Of course, it's much more sophisticated than that! But it is a good analogy, because it emphasizes the two characteristic parts of the system: A source of heat produced by electricity through some sort of a resistor, and a source of (typically non-reactive) fluid.
Resistojets have been flown in space, and do well in situations where energy is much more plentiful than mass, and where propulsion efficiency needs to be reasonably high but low-thrust is acceptable.
Arcjet
Arcjets are a form of electric propulsion, whereby an electrical discharge (arc) is created in a flow of propellant (typically hydrazine or ammonia). This imparts additional energy to the propellant, so that one can extract more work out of each kilogram of propellant, at the expense of increased power consumption and (usually) higher cost. Also, the thrust levels available from typically used arcjet engines are very low compared with chemical engines.
Hall Effect Thrusters (HETs)
Also known simply as plasma thrusters, HETs use the Hall Effect to accelerate ions to produce thrust. A varient called stationary plasma thruster (SPT) has been used by the Russians for stationkeeping for many years and will be used on Western satellites soon.
One problem with HETs is that their plumes diverge quickly and thus can impinge on other spacecraft parts leading to thermal and contamination problems.
Ion Thruster
Of all the electric thrusters, ion engines have been the most seriously considered commercially and academically. Ion engines use accelerated beams of ions for propulsion.
Ion engines are best used for missions requiring very high delta-V's (the overall change in velocity, taken as a single value), interplanetary missions, for example. This is because the more performance required of the propulsion system, the faster a high efficiency system like ion engines will pay off.
NASA has developed an ion engine called NSTAR for use in their interplanetary missions. Hughes has developed the XIPS (Xenon Ion Propulsion System) for performing stationkeeping on geosynchronous satellites.
Magnetoplasmadynamic (MPD) Thrusters
Thrusters that use the Lorentz force (a force exerted on charge particles by magnetic and electrical fields in combination) are called magnetoplasmadynamic (or MPD) thrusters. MPD thruster technology has been explored academically, but commercial interest has been low.
MPD thrusters can be run in a steady state fashion or in a pulsed mode.
Pulsed plasma thruster (PPT)
Pulsed plasma thrusters use an arc of electric current through a solid propellant (almost always teflon), to produce a quick and highly efficient and dependable burst of impulse. PPT's are great for attitude control, and for main propulsion on particularly small spacecraft (those in the hundred-kilogram or less category).
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