Orion1
At the risk of over simplifying,
Power is generally produced by an engine that turns a propeller ie not a jet engine. The power produced by the engine is of no use to you until you slap a propeller onto it - the propeller/engine combination then produce thrust.
The thrust required to fly an aircraft exactly matches the total drag curve(the bucket graph), so thrust is a fairly simple concept.
The power required to achieve the above is not so simple - it varies with the thrust required to fly(bucket graph), as well as with the speed at which you are flying.
In order to determine a power requirement at a given speed, you go to the bucket graph and extract the thrust requirement and then multiply it by the TAS. This will give you the power required to fly at that speed. Note : this is a hypothetical exercise. It is not intended for you to actually calculate.
In terms of climb performance, it is generally accepted that best ROC will be where the most excess power is. I am sure that there are ways to mathematically prove that best AOC will occur at a certain point on the power required/available curve(I think it generally co-incides with V for min power), but I think the boffins in the white coats thought it simpler to explain AOC to us in terms of thrust rather.
Anyway hope it helps.