The ECU or Engine Control Unit controls all aspects and function of your cars engine. Pretty much every car since 1992 have an engine management unit controlling at the very least fuel/timing.
Modern demands for fuel economy and strignent emissions requirements the computer has evolved into a complex system. 
Pre ECU driven engines had timing managed by a crude mechanical means which advanced timing as the revs increased. This was always a compromise at best but did the job well.
The ECU contains a complete map which not only set the spark timing according to engine speed, it also controls lots of other things. On a turbo engine boost is controlled, fuelling and these can be altered depending on load, throttle position and engine and air temperatures.
They enable an engine to be run at peak efficiency and will adjust or trim engine parameters to protect the engine from damage in most situtations encountered like poor quality fuel, or high engine temperatures. Obviously extreme problems cannot be circumvented but engines are more reliable today as a result.
For example the air fed into the engine with a calculation on air temperature, the amount of unburnt O2 in the exhaust allow pretty precise timing alterations to be made.
Standard maps are designed to cope with varying emissions laws around the world and widely varying fuel quality and running conditions. So you can appreciate they are usually a very safe fudge at best.
If you are willing to run closer to the edge of reliability or will maintain your car with a shorter service intervals there are massive potential gains from a remap.
A diesel engine will usually become more economical after a remap and will offer almost silly levels of increased torque. Petrol engines with turbos offer large power gains but the fuel economy is always lower.
On most turbocharged cars there are substantial power increases on offer. (N.A.S.P engines will not really benefit from a remap unless a large number of other modifications have been done first or the manufacturer have detuned the power output.)
Off the shelf remaps are very general and take the OEM state of tune up a notch or two, these have a place and can make a massive difference.
Custom remaps are engine specific, and as no 2 engines are the same it's generally offering larger power gains and better drivability than the off the shelf remap.
Where other mods have been done, such as the fitting of a CAM or uprating the turbo, a custom remap is an essential. New switchable remaps are avialable which allow the driver to select from an economy or valet mode to a sports/track day mode.
Some companies offer handheld remapping units which contain a selection of maps and can be flashed at will. We tried the ByteFlash unit which worked very well.
There are also a few maps around which can be switched on and off from inside the car via a switch or dash control, these are quite neat solutions but can cost a lot more.