Diesel engines, like the gasoline engines you are familiar with, are internal combustion engines. Much like gasoline engines, diesel engines run by injecting some sort of fuel above a cylinder and igniting it to cause an explosion, pushing the cylinder down. The downward motion of the cylinder turns a crank which turns the wheels of the car.
Diesel engines differ from gasoline engines because they use the piston to compress air only (not an air/gas mixture) and use spontaneous combustion instead of a spark to ignite the fuel. This allows diesel engines to use a higher compression ratio, and therefore work more efficiently, potentially leading to environmental benefits.
Diesel engines are already widely used. About 25% of cars in Europe are diesel. Current diesel technology allows diesel engines to be 25% more efficient (that is, use 25% less fuel) than gas engines and emit 20% less CO2. On the other had, NO, NO2 and particulate matter emissions tend to be higher, although new technologies are reducing this problem.
Diesel engines tend to be significantly more expensive than gasoline engines but also last longer, as they are built to withstand higher pressures and temperatures. Diesel engines also have more torque (power) but less top speed/acceleration than gasoline engines.
It is interesting to note that the first diesel engine, by Rudolph Diesel in 1900, ran on peanut oil.