How they work.
When the key is turned on the ECU runs through a quick self-check that tells it everything is ready to go.
As the engine spins, the crankshaft position sensor tells the ECU how fast the engine is turning, where TDC is and what position the crank is in, while the throttle position sensor and MAP sensor indicate load. Note that both Alpha-N and Speed Density systems employ Throttle Position Sensor or TPS and MAP sensors. However, how the information is interpreted depends on the type of mapping being used.
The ECU compares the supplied info to its stored map, and selects a value it likes, that value representing the amount of time it'll hold the injector open, (its pulse width). If the exact value can't be found the computer will estimate what the engine needs by searching the map for higher and lower numbers and then picking something in the middle.
Once it has a value it likes, the computer will check to see if any further adjustment is needed. For example, if the engine temperature sensor tells the ECU the engine is stone cold or if the ambient air temperature indicates that the incoming air is particularly chilly ECU will assume the engine is in a cold start situation and select a fuel point that's rich enough to fire the engine, depending on the systems level of sophistication it may also increase the idle speed by moving the throttle stop via a small electric motor.
As the engine warms up and less fuel is required, the sensors transmit that info to the ECU, which will then lean out the mixture and lower the idle.
Obviously that's a very simple explanation of how an EFI system operates, and glosses over the literally thousands of inputs, calculations and decisions the ECU makes in fractions of second, but essentially that's how they work, everything else is just a detail.
System Overlap
Since both Speed-Density and Alpha-N systems share common sensors, and hold distinct advantages over each other at different throttle positions it's not uncommon for a manufacturer to piggy back the two systems together. Many EFI's systems use a Speed-density map at low and cruising speeds where the vacuum system is strong, then switch to an Alpha-N strategy at large throttle openings. In essence, this provides the best of both worlds, which is always a neat trick.
Open/Closed Loop Operaton
The ability to adjust itself, based on mapping is EFI's biggest advantage over a carburetor so any discussion of it has to include some comment on open and closed loop operation. A system that's operating in open loop is using the information it receives from its sensors to select a predetermined amount of fuel from its stored map. Once it selects the air/fuel ratio it wants it doesn't particularly care what happens in the combustion chamber or what's coming out of the tail pipe. The majority of fuel-injected motorcycles operate with open loop systems.
A closed loop system is one that uses an oxygen sensor to measure the amount of unburned oxygen in the exhaust system and adjust the mixture based on its inputs. For street use oxygen sensors are primarily used to limit emissions at idle and cruise speeds, and to protect the catalytic convertor (if one is being used) from overdosing on fuel. Closed loop systems operate only when there's little demand for power and conditions don't change very quickly, I.E. during idling and cruising. Whenever power is called for, the system must revert to an open loop. Because they cause the engine to run lean, closed loop operation can also cause surging problems at light throttle openings, especially on twin cylinder engines. For street use, closed loop systems are primarily driven by emission control requirements, so I'm positive we'll see a lot more of them in the future.