A fuel injector consists of three main components: a solenoid, a needle valve, and a nozzle, which breaks the stream of fuel released into a cloud of fine droplets.
Two major types of fuel injection system exist: Central Fuel Injection, and Electronic Fuel Injection. On a CFI system, there is one injector which is used in place of the valves and jets in a carburetor. EFI puts one injector at the intake valve to each cylinder.
The fuel/air mixture can be precisely controlled by the system's ECU, a small embedded computer, by rapidly pulsing power to the solenoid on and off. The solenoid pulls back on a needle valve inside the injector body, opening the valve. A return spring closes the valve when the circuit is de-energized. Opposite the valve is the nipple which connects the injector to the fuel rail, a metal tube which distributes fuel at high pressure to each injector.
Next to your fuel injectors on the fuel rail is a pressure regulator. The pressure is variable and controlled by an amount of engine vacuum allowed to pull through an air valve, also controlled by the ECU.
Injectors can, and sometimes do, fail. The nozzle can become clogged with goo over time, which can cause it to misfire. The solenoid's coils can also become open or shorted, causing the injector not to activate at all. The resistance of the solenoid can be checked with a multimeter and compared against the acceptable reading found in a service manual, or that of a known good injector.
WARNING: If you're going to remove or otherwise mess with the injectors, discharge the pressure on the fuel rail first! There will be a Schrader valve near one end for this purpose. Wrap a rag around a small screwdriver, press the rag down around the valve, and press the core in with your screwdriver to release the pressure. Make sure the ignition cannot be turned on while you are working on the fuel injection system. Pull the battery cable! Also, replace the injector seals each time you remove and reinstall them. Trust me, you don't want fuel shooting out at high pressure.