The OP's question is a fair one IMO. A lot of people talk about mapping on here but very few describe what it is. It would be a difficult question to search for on here because the phrase is used so much, but almost always with little/no explanation.
I know very little about it, but here goes: an ECU contains information about how to manage the engine (how much fuel to inject into the cylinders, etc etc) for a given set of input parameters (engine revs, engine temperature, possibly things like boost pressure). In short, it contains a map of engine management vs inputs. The range of inputs in a standard map doesn't go from "0 to infinity": they only cover a small-ish finite range that covers all the likely input values. The map provided by Toyota is fine as long as the inputs aren't going to fall outside that range. When they do fall outside the range (i.e. as a result of major performance modifications) then the ECU needs to be re-mapped in order to cope with the new range(s). Toyot'a stock ECU can't be re-programmed to hold a different map, so a different ECU is needed if a car is to be re-mapped.
This is a noddy explanation which leaves many holes and it might be incorrect in some details, but hopefully it'll give the OP a foothold on what we all mean by a re-map.