Solid-fuel rocket:
Solid fuel rockets are the oldest kind of rocket propulsion, invented in China in the 13th century. Solid rockets use solid fuel and oxidizer instead of liquid fuels, which allows them to be stored easily for long periods of time. However, they are less efficient, and cannot be turned off after ignition and before they run out of fuel. These attributes make them popular for military applications because of their ability to be stored for long periods of time, and as booster rockets for the first stage of liquid rockets because of their low cost.
Hybrid rocket:
These are a mix of solid fuel rockets and liquid rockets. In a hybrid rocket, one part of the fuel (either the fuel or the oxidizer) is in a solid form, similar to a solid rocket booster, except it cannot burn by itself. The other part of the fuel/oxidizer mix is stored as a liquid in a pressurized tank attached at the bottom to the top of the solid part. When the valve is opened from the tank, and the mix is ignited, the fuel and oxidizer burns. Its advantages over solid rockets are mostly in safety, because that the engine can be quickly shut down. Also, no turbomachinery is required.
Monopropellent rocket:
Monopropellent rockets use a single fuel by bringing it into contact with another chemical to produce a reaction which adds energy to the fuel. Wikipedia has a good explanation of the different kinds of chemical reactions. They are popular as reaction control system rockets, because of their simplicity and controllablity. However, they have very low specific impulse, so they are poor primary engines.
Hypergolic rocket:
These work much like normal liquid fuel rockets, except the fuels and oxidizers ignite on contact. That's great for ease of storing, since no cryogenic tanks are necessary, but they are incredibly corrosive, toxic, and carcinogenic. Even my spell-checker doesn't like them. "Hyperbolic rockets?"
Nuclear thermal rocket (NTR):
In a nuclear thermal rocket, a fuel (probably hydrogen) is pumped around a nuclear reactor so it expands, and then expelled out the rocket nozzle. It would have a specific impulse of about 850s, which is considerably higher than more conventional kinds of rocket engine. It wouldn't be as powerful as a chemical engine (thrust-to-weight (TWR) ratio of 7:1 instead of 70:1) but it would excel in transfer stages where TWR is less important. A lot of Mars mission proposals involve NTRs in some way, however, their development has been slowed since the cancellation of project rover, because of (very well-founded) environmental concerns over open air testing of nuclear rockets:
Ion rocket:
These work by accelerating plasma (or ions) for use as a fuel. They get extraordinarily high specific impulse, but at the cost of very low TWR, on some engines, equivalent to the weight of one sheet of paper constantly accelerating the spacecraft. Honestly, I have no idea of the specifics of how they work, so take a look at Wikipedia for a detailed explanation. However, I do know that they use a unusual fuel, such as xenon, and ionize it with electricity, propelling it outwards at 20-50 kilometers a second.
Of course, there are lots of kinds of more exotic rockets, some extremely theoretical, with names like Pulsed plasma jet, Fission sail, Nuclear pulse propulsion, reactionless drive, microwave powered rocket, Bussard ramjet. Project Rho has a nice series of articles on these almost-sci-fi propulsion methods.
