For the layman.
The cost index is a value which represents a ratio between least fuel and least time for a specific flight or part of flight. It is a method of reducing or rather, optimising the overall costs of a specific flight or series of flights.
The least fuel will be burned by flying at the maximum range speed. This is often quite slow. It may be compromised by the available cruise altitudes.
The least time will be used by flying at max speed. It may also be compromised by available altitudes.
For the cheapest overall flight costs there will be an optimum ratio of time/fuel.
When fuel is cheap the ratio will be biased towards higher speed.
When fixed costs are cheap the ratio will be biased towards lower speed.
The actual speed will vary with weight and altitude.
It will also need correcting for head or tail wind. Head wind is like making fixed costs higher as it takes longer for the flight therefore it will move the optimum ratio towards a higher speed.
The implementation of the index value is different on different aircraft. Some will allow the crew to enter in directly into the flight management computer system. This will then calculate the optimum speeds for the planned climb, cruise and descent phases of the flight. Other systems will only calculate the cruise speed and some systems will require the crew to calculate the appropriate speed via tables or graphs and either enter this value into the the flight management system or even select and fly the speed manually.
The concept is sound in that if correctly implemented it should reduce overall operating costs but like all things, if you start with the wrong data, for example real fixed costs, you will end with the wrong speeds and end up increasing costs rather than reducing them.