The Definition of Power

Power
POWER is defined as an amount of work (foot-pounds) done in a given amount of time (seconds or minutes). Power is given in foot-pounds-per-minute, and is calculated by the following formula:

Power (P) = Work foot-pounds)/Time (seconds or Minutes) 
                  = foot-pounds/seconds or minutes 

If the work is computed in inch-pounds, it must be divided by 12 to convert it to foot-pounds.

For the amount of power to be meaningful, it must be compared with some unit of measure. The common unit of power measurement is the horsepower, which is expressed as follows:

1 Horsepower = 33000 foot-pounds/one minute or 550 foot-pounds/one second

In order to determine the pump horsepower, you have to calculate the horsepower required on the cylinder end. 

To do work or use power, energy must be expended. The "Law of Conservation of Energy" state that "Energy cannot be created or destroyed: It can only be transformed."
As you know, not all energy is used to perform work. A certain amount of energy is expended, when doing work, to overcome friction. This energy is not lost, but change into heat energy.
The types of energy used in hydraulic systems include the following:

ELECTRICAL ENERGY => to operate the pump motor.

HYDRAULIC ENERGY => produced by pump

KINETIC ENERGY => produced when the hydraulic fluid moves a piston.

POTENTIAL ENERGY => produced when the piston has raised an object.

HEAT ENERGY => produced by friction in the pump motor, pump, piston and hydraulic fluid.

Incompressibility and Non diffusion
One of the problem encountered in a hydraulic system is that of storing the liquid. Unlike air, which is readily compressible and is capable of being stored in large quantities in relatively small containers, fluid cannot be compressed. If you want to store large amount of hydraulic fluid in small tank, it is not possible because liquids are INCOMPRESSIBLE. (Liquids can be compressed slightly, but only in such a small amount that it is usually not considered.)

DIFFUSION can be described as the rapid intermingling of molecules of one gas or liquid with another. This should not be confused with evaporation, which is the changing of a liquid to a gas. Because of its slow evaporation rate Hydraulic fluid can be placed in an open container or poured from one container into another without diffusing. This is NON DIFFUSION. Gas, however, cannioot be placed in open containers, because they would rapidly diffuse into surrounding air. Therefore, gases ar always stored in closed container.