You achieve high pressure diesel injection when you
inject diesel fuel under a very
high pressure in the
cylinders of an
internal combustion engine.
High pressure injection yields smaller drops of fuel, which have relatively more surface exposed to the air surrounding them. Thus, the oxidation (the burning process) of the diesel is performed faster and better. This means that you get more energy out of your fuel, and exhaust particles that are smaller.
The overall result are engines that generally run better, and yield more torque and power. As a bonus, they are far more efficient, lowering fuel use and thus improving milage. That is, because the fuel is burnt more thoroughly, you get more energy out of it.
Normally, diesel engines like this would produce way too much noise to be usable in passenger cars. But the injection system has a clever way to deal with it. In gasoline engines, the ignition of the fuel is done by an electrically generated spark. Not so in diesel engines, where the burning process starts automatically due to the high temperature and pressure in the cylinder. So it seems impossible to control this process! Of course it is not: the motor management decides exactly when the fuel enters the cylinder. And by gradually injecting the fuel (not everything at once), the burning process can be extended over a (slightly) longer period. This reduces the noise that the combustion produces, and lowers the exerted force on the components of the engine, thereby improving the durability.
One drawback that has been coined against high-pressure diesel engines, is that the smaller particles they generate are more likely to be inhaled directly into the lungs of people and animals, where they can cause considerable damage (*cough* cancer *cough*).
Another one is that you need clean fuel. At these pressures and quantities, one cannot tolerate impurities of the diesel, and the chemical characteristics must be constant. As a result, e.g. in Africa you'll probably want to use a less advanced engine, just a simple workhorse that even runs on polluted, low-quality diesel. In America, some of the highest-pression diesel engines for family cars (e.g Volkswagen's) are not being sold, because the American Diesel is less pure than e.g. the European (being more similar to the 'industrial' Diesel for trucks).
For examples of high pressure diesel injection systems, see what Volkswagen does with its pump injection system. Other car manufacturers use common rail.