Hmmmm .....
Well, the first thing you need to know is about Aircraft Performance. This is a complicated subject, and represents one or more of the exams that pilots have to sit for their professional pilot licences. This tells you what an aircraft is supposed to do, performance-wise, when it takes off.
Aircraft performance at take-off relates to: amount of runway used if take-off is continued; amount of runway used if take-off is stopped with an engine failure at a critical speed; amount of ground covered to clear an obstacle that is 50' above the height of the runway (or 35' if wet); ensuring a climb gradient that clears obstacles in the take-off path by a given amount, again assuming an engine has failed; use of anti-icing and bleed systems, consideration of runway gradient, air temperature, pressure, wind and so on.
What you come up with is an RTOW - regulated take off weight. This is the maximum weight of the aircraft that will allow it to comply with all these legal requirements.
However, the aircraft will often be operated at considerably less than this RTOW. In the case of both B737 and A320, "derating" (on the day) operates by determining the highest temperature at which the aircraft could take off at and telling the engines to pretend that this is the actual temperature (which commands a lower power setting). On the A320/321, the temperature is called the "flex" temperature; on the B737, the "assumed temperature."
There is also a difference between max power, which is typically used at most for short bursts (at the start of a go-around, or full-power take off), max continuous power, which is typically used for single engine work, and climb power which is the normal "full power" away from the ground.
In short, it's hideously complicated and technical - I hope my answers aren't too misleading! I expect someone will correct me if they are.