Remapping is essentially moding map which is stored in car's ECU. Map in ECU is controlling how the engine works. How much fuel,air and so on.. is given at the exact moment. Map is divided into sectors which control different functions. For example: first sector is for how much fuel is injected, second is for how much air is injected, thrid is for timing the spark plug firing... There are much more sectors which must be modified to achieve anything. By modifying those sectors you can change the quantity (fuel, air) or timing (spark plug glow) which is injected into car's engine.
In short, you gain or lose horsepower and torque (performance) depending what you want, also in many cases you will get better fuel economy but you will probably shorten your engine lifetime (that depends how heavily map is modified).
Also worth mentioning, car remapping can be divided into stages. For example, Stage 1 is map modifying only, Stage 2 is more aggressive map modifying and maybe fitting performance parts such as performance air filter or more powerful fuel pump, Stage 3 requires even more moding...
Some of the things you can turn off are: Vmax (max speed limiter), REVmax (max rev limiter), DPF off, EGR off...
By reading your post about the car problems I would advise you not to do any remapping on the car until you fixed all problems and errors. Car's engine and transmission should be in excellent condition before remapping.