It comes down to how the individual systems work and what the goal of adding more power is.
In some cases, such as with life support as @jwenting mentioned, the extra power may simply be to ensure continued operation of the system if primary and standard secondary sources of power are interrupted.
With weapons the extra power would be to increase their destructive power or place them in an overloaded state, as @apoorv020 said.
Extra power to shields is used to maintain the shields when the standard capacitance of the shield generators is reduced to 0 and to restore their full power more quickly when not under fire.
With engines the effect is similar to weapons, but with the goal of increasing their output for more speed (think of reving the engine in your car in to the red zone).
In most cases routing additional power to a system for extended periods is probably not a recommended action. I would also bet that after the episode the engineers probably pull a couple extra shifts to check for and repair any damage caused by minor overloads. But in a pinch routing additional power to one system or another can turn the tide in a battle, get you somewhere a little bit sooner or increase the "grip" of your tractor beam to move that large asteroid.