Remapping the beginning.The Ecu,the engines Brain.

The ECU is basically a small computer that carries out all the controls of the Engine. From engine spark advance, to controlling the turbo wastegate, Air and Fuel metering and many many more variables and controllers .
Back in the good old days, the task of timing the ignition spark was performed by the distributor. The greater the RPM, the more the timing would advance. It was clunky and mechanical but seemed to work quote well. This did a reasonable job, but for the most effective power you would need to vary the timing to a greater degree than a fixed ratio advance curve.
The electronic ignition system was born giving much finer control over fuel delivery and spark timing.
A complete map of variables was entered into the ignition program and preset timing would be read from a table. Now air temperature, engine speed, engine load and even control over turbo/wastegate control & fuel delivery rates means that precise management of the engine ignition timing is possible and you can achieve the maximum power output throughout the rev range.
So what does remapping do?

Electronic ignition allows the manufacturer to fine tune economy at popular road speeds such as 50kmh, 80kmh and 100kmh where most cars spend a large proportion of their time. It is now possible to advance the timing if the throttle is wide open to give greater power or back off the timing when cruising at constant speed. When a manufacturer creates a timing map they build into it a big margin of error to cope with: adverse temperature ranges, minor faults & bad conditions.
In some countries emissions targets vary as measured by CO2, HC, NOx and these usually require a one setting fits all approach. Manufacturers do not want people breaking down, suffering premature parts failure so they build in a wide margin of tolerance. Different countries use different grades of fuel and have varying degrees of extreme weather conditions, all these factors add to the one setting fits all that has to be done to keep all the cars working well across the globe. Each car that leaves the production line is also unique, some achieve 10bhp less and others can be 10bhp up on standard specs, depending on how well the components are machined and put together. So rather than put each car through a unique assessment and get a bespoke timing map, they adopt a standard one map fits all philosophy.
It is also a fact that manufacturers use the remap to produce different power versions of the same engine and get lower insurance cover ratings and better fuel consumption.
You start to see the fantastic scope for improvement, when you add into the mix the fact that a lot of vehicle owners will be adding better performing components to the car, you have a really strong case for a remap.