Diesel engines were among the first to use pressurised injection of any sort, and have (for the most part) stuck with it ever since. Fuel injection differs from carburetion in that it uses fuel pressure to spray engine into the engine, as opposed to carburettors that basically "leak" fuel into the engine using vacuum. There are three basic types of valves (injectors) used to vent fuel pressure from the lines into the engine's intake or cylinders.
Spring-loaded (aka "mechanical") injectors are the oldest type of fuel injector, and are still commonly used today in many industrial applications. Diesels long ago adopted to innovations that demanded very high fuel pressures and quick injection: turbocharging and direct cylinder injection. Both of these force the injector to operate in very high-pressure environments, which would force air back into the injector if not for those very high fuel pressures. Mechanical injectors have very quick-acting spring valves inside them. Once the fuel pump supplies enough pressure for that cylinder's injector, the spring valve snaps open and squirts fuel into the engine. These injectors ensure that fuel always comes out at the same pressure, with the same timing and frequency. Without these spring valves, the rising and falling fuel pressures would "trickle" fuel into the engine rather than squirt it.
The solenoid injectors used on diesels are largely identical to those used on gas engines. Solenoid injectors use a set of electromagnets to open the valve; when the computer sends electricity to the injector, the magnets energise and pull the injector valve away from the valve seat. When the magnet turns off, a small spring shuts the valve.
Piezoelectricity is an amazing but little-known electromechanical phenomena. Piezoelectric materials can change shape when electricity is applied, or can emit electricity when subjected to sudden force. Many very common materials exhibit some amount of piezoelectricity, including silk, cane sugar, quartz and dry bone. Hit a piece of silk on an anvil with a hammer and it will actually produce a small but measurable electric current.
Piezoelectric injectors work on the opposite principle; electricity applied to the crystal or ceramic inside the injector causes it to expand slightly. This expansion opens the injector valve, allowing it to spray fuel. Piezoelectric injectors can open and close very quickly and are among the most accurate available.
Introduced by Caterpillar Diesel, the HEUI (Hydraulically Actuated, Electronically Controlled, Unit Injection) uses oil pressure to press on a diaphragm inside the injector. This diaphragm pushes on the small amount of fuel inside the injector, pressurising it to the massive 3,000 to 21,000 psi required for direct injection. Because the injectors themselves act to pressurise the fuel, HEUI systems can make do without the powerful fuel pumps that make other systems heavy, expensive, dangerous and difficult to control. This is a huge boon for computer-controlled common rail diesels; that pressurised fuel rail is essentially a bomb waiting to go off.