Electronic ignition is the control of ignition of the fuel/air mixture in an internal combustion engine by use of electronic rather than simple electromechanical means. In its simplest form it provides no benefit other than durability, but more advanced designs allow vehicles to run more efficiently.
Until recently, the most common ignition system for a car was to have a distributor driven directly from the crankshaft such that for every two revolutions the crank made, a rotor in the distributor made one. This is because in a four stroke engine, combustion occurs in each cylinder on every other cycle. The rotor would cross a contact in the distributor, thus completing a circuit which would send the coil voltage to the spark plug, hopefully causing the fuel/air mixture in the cylinder to combust.
Timing advance was made possible in these systems by the use of a vacuum advance system. When the engine is running it draws air into each cylinder once per two revolutions; this creates vacuum, which can be used to power various systems around the car. As the vacuum increased, the timing was advanced further, because as RPMs increase, you need to advance timing to cause the fuel to achieve full burn just after the piston begins descending. Too soon, and you get detonation which will eventually destroy your engine; too late, and you lose efficiency.
The really oddball part of the system is explained when you discuss points, or "breaker points". A cam on the distributor shaft opens and closes the points in order to cause the coil to generate the high-voltage pulse needed to fire a spark plug. This pulse comes when the rotor is near the contact on the inside of the distributor cap.
The problem with a system like this is that it cannot accomodate any changes in its environment. Operating at a greater or lower altitude or otherwise changing the atmospheric pressure of the working environment will significantly change the behavior of the engine. Similarly, a change in octane level will change the point at which the fuel combusts, changing the necessary timing. This is one reason older cars run signifcantly better or worse on different brands of gasoline, or even gasoline from different service stations of the same brand.
Also significantly, the engine is completely driven by the amount of fuel you are delivering. The carburetor has only a few main features, and perhaps the most significant are a flap which controls the amount of air let into the engine, a valve through which fuel delivery is controlled, and a nozzle (called a "jet") which turns the stream of fuel into a spray. When you press the pedal down further, the flap opens to admit more air, the fuel delivery is increased, and everything else has to keep up.
What, then, is electronic ignition? It is the use of a crank angle sensor to determine the position of the crank, and adjust the engine's operation accordingly. An electronic ignition system typically uses two methods to achieve this. The first is controlling the duty cycle of the fuel injectors (if present) to vary the rate of fuel delivery. The second is to advance or retard the timing. The ignition system will generate the pulse from the coil at the proper time based on the position of the crank angle sensor rather than using points. This can be done with a very simple computer, as all it has to do is control the timing advance, and the firing of the coil. It is not strictly necessary to use a microprocessor at all for basic operation, and even for more complicated systems a simple microcontroller will usually suffice.
Crank angle sensors basically come in two types: optical, and magnetic. Both work through the use of a transmitter and a pickup, which are interrupted by something passing between them, producing a square wave output. In an optical sensor, an infrared LED shines onto an IR sensor, and is interrupted by a disc (usually metal) with holes in it. In the magnetic type, a magnet is on one side, and the pickup is either a coil or a hall effect sensor. A diode is used to convert the coil output into a crude approximation of a square wave; as metal vanes of a reluctor wheel pass between the magnet and the coil, a magnetic field is interrupted and voltage is produced. Hall effect sensors are solid-state, and produce a square wave as the reluctor's vanes pass. Optical sensors typically have higher resolution than the magnetic; Magnetic sensors usually trip only once per revolution for each cylinder, while optical sensors often put out a signal for each degree of crank rotation - there are 360 holes in the disc.
The primary benefit of a system of this type is reliability; there are no points to wear out, requiring replacement.
Enter the age of fuel injection. A fuel injector is just a solenoid-controlled valve with a restrictor which allows a very specific rate of flow at a range of pressures, and a spray nozzle which "atomizes" the fuel as it enters the engine. By switching them on and off very quickly it is possible to spray metered amounts of fuel into the combustion chambers (directly, in the case of direct fuel injection, or through a manifold, in the case of throttle body fuel injection) in order to control the amount of fuel used.
However, that is by no means the entire picture. The fuel injectors are controlled by an onboard computer which is tied into the engine via a number of sensors which can monitor different operating characteristics. The ECU typically knows a fair number of things about the way the car is operating, including RPMs, crank angle, a measurement of O2 in the exhaust gases, and so on. The best systems also have a knock sensor to detect detonation, and a mass air flow/MAF or mass air pressure/MAP sensor to determine the amount of airflow into the engine, as well as an intake air temperature sensor. The more information you have about your operating environment, the closer you can get to a stoichiometric ratio of air to fuel, which is the point at which all fuel is consumed and also the point of highest efficiency.
Generally speaking, in a car with electronic fuel injection (EFI), the pedal controls only the flap admitting air to the engine, at the front of the throttle body. It is also coupled to a throttle position sensor which lets the car know how much you're pressing down the pedal.
How does this tie into electronic ignition? In a vehicle using electronic fuel injection, everything is controlled by the ECU, or engine control unit; timing advance, fuel delivery, the firing of the coil. The only things generally not controlled by the ECU are the intake valve aperture (the aforementioned flap controlling air intake) and the distributor position. The primary goal of the ECU is to do what the user wants (IE, deliver fuel based on pedal position) while keeping the O2 levels as low as possible, indicating that as much fuel is being burned as possible. If present, the knock sensor is a sign to the ECU that it should retard timing, as detonation is the sound of your piston trying to move sideways in the cylinder due to premature combustion.
Finally, the airflow into the engine has a significant effect on what your timing advance should be. There are two systems used by engine management systems to determine airflow: speed density, which assumes so many CFM for a given speed; and mass air which actually measures airflow. The latter also has two subcategories: mass air flow, in which the speed of incoming air is measures, and mass air pressure, in which the pressure in the intake is measured. Mass air pressure is the best of the possible solutions because it will compensate for forced induction and unusually high or low atmospheric pressure.
If you're going to go through all of this trouble to remove troublesome moving parts, you can do away with even more of them if you eliminate the distributor. This eliminates the distributor shaft (as well as the gears at its base that drive it from the crankshaft), the rotor, and of course the distributor itself. It also eliminates all hardware which deals with the timing advance, as this can now be done on the ECU in software.
There are two ways to do this. The first is to mimic the behavior and form of the distributor. This provides significant advantages in reliability, but is hardly the best way to go about this. Generally speaking this is done with a waste spark system in which there is a coil for each two cylinders. The coil is fired in one direction for one cylinder, and the other for the other cylinder served by the coil, which is opposite the firing cylinder in the firing order. Both plugs spark, but the non-firing cylinder's spark is "wasted" as it does not cause ignition.
Instead, the automobile industry is moving towards coil-on-plug ignition (COP). Rather than using one large coil to drive all of the spark plugs, each plug has its own smaller coil, usually mounted directly to the plug, and a low-voltage trigger wire running to the ECU. This eliminates the coil wire and all spark plug wires, which are somewhat prone to failure due to the harsh environment under your car's hood. It also means that each individual ignition coil does less work, as it only has to service one spark plug, and that if one coil should fail you can still limp home on the remaining firing cylinders. The car can generally detect this behavior and stop delivering fuel to the "dead" cylinder as well.
There are a number of aftermarket kits to add electronic ignition to older cars. The most notable are from accel (which makes cheap but adequate systems suitable for daily use) and MSD, which makes a part suitable for racing. It is generally paired with a larger coil and fires each spark plug multiple times in order to achieve a more complete combustion of available fuel.
Generally speaking, the installation of a system like this is fairly simple, and involves the addition of a hall-effect sensor and a magnet to the distributor, and replacement of the coil and sometimes replacement or augmentation of the timing advance system. In the case of vehicles with points, the distributor is often replaced entirely. The magnet, when paired with the hall-effect sensor, provides a low-resolution crank angle sensor which is then used by the ignition system to control coil timing.